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MODULARITY OF GENERATING SERIES OF DIVISORS

ON UNITARY SHIMURA VARIETIES

JAN BRUINIER, BENJAMIN HOWARD, STEPHEN S. KUDLA, MICHAELRAPOPORT, TONGHAI YANG

Abstract. We form generating series of special divisors, valued in theChow group and in the arithmetic Chow group, on the compactifiedintegral model of a Shimura variety associated to a unitary group ofsignature pn´ 1, 1q, and prove their modularity. The main ingredient ofthe proof is the calculation of the vertical components appearing in thedivisor of a Borcherds product on the integral model.

Contents

1. Introduction 12. Unitary Shimura varieties 133. Toroidal compactification 374. Classical modular forms 615. Unitary Borcherds products 646. Calculation of the Borcherds product divisor 767. Modularity of the generating series 888. Appendix: some technical calculations 99References 109

1. Introduction

The goal of this paper is to prove the modularity of a generating series ofdivisors on the integral model of a Shimura variety associated to a unitarygroup of signature pn´ 1, 1q.

This generating series is an arithmetic analogue of the classical thetakernel used to lift modular forms from Up2q and Upnq. In a similar vein,our modular generating series can be used to define a lift from classicalcuspidal modular forms of weight n to the codimension one Chow group ofthe unitary Shimura variety.

2000 Mathematics Subject Classification. 14G35, 11F55, 11F27.J.B. was partially supported by DFG grant BR-2163/4-2. B.H. was supported in part

by NSF grant DMS-0901753. M.R. is supported by a grant from the Deutsche Forschungs-gemeinschaft through the grant SFB/TR 45. S.K. is supported by an NSERC DiscoveryGrant, T.Y. was partially supported by NSF grant DMS-1500743.

1

2 J. BRUINIER, B. HOWARD, S. KUDLA, M. RAPOPORT, T. YANG

1.1. Background. The construction of modular generating series whosecoefficients are geometric cycles begins with the work of Hirzebruch-Zagier[HZ76], who considered the cohomology classes of divisors T pmq on com-pactified Hilbert modular surfaces over C.

An extensive study of the modularity of generating series for cohomologyclasses of special cycles in Riemannian locally symmetric spaces S “ ΓzXwas undertaken by in a series of papers [KM86, KM87, KM90] by the thirdauthor and John Millson. The main technical tool was a family of Siegeltype theta series valued in the deRham complex of S, from which modularitywas inherited by the image in cohomology.

The special cycles involved are given by an explicit geometric construc-tion, and so, in the cases where S is (the set of complex points of) a Shimuravariety, it is natural to ask whether the analogous generating series for spe-cial cycle classes in the Chow group is likewise modular. In the case ofShimura varieties of orthogonal type, this question was raised in [Kud97a].Of course, up to (often non-trivial) issues about compactifications, the mod-ularity of the image of such a series under the cycle class map to cohomologyfollows from the work of Kudla-Millson. Indeed, in some cases, this alreadyimplies the modularity of the Chow group generating series. See, for exam-ple, [Zag85, YZZ09].

The generating series for the images of Heegner points in the Jacobian of amodular curve was proved to be modular by Gross-Kohnen-Zagier [GKZ87].Motivated by their work, Borcherds [Bor99] proved the modularity of thegenerating series for Chow groups classes of Heegner (=special) divisors onShimura varieties of orthogonal type. His method depended on the miracu-lous construction of a family of meromorphic modular forms on such varietiesvia a regularized theta lift [Bor98], whose explicitly known divisors provideenough relations among such classes to prove modularity.

The passage to cycles on integral models of Shimura varieties and thegenerating series for their classes in the arithmetic Chow groups was initiatedin [Kud97b]. In the case of special divisors on Shimura curves, or, moregenerally, Shimura varieties of orthogonal type, the required Green functionsconstructed explicitly there are derived from the KM theta series. Quitecomplete results on the modularity properties of generating series for specialcycles on Shimura curves were obtained in the book [KRY06]. There thecase of arithmetic 0-cycles is also treated and the corresponding generatingseries is shown to coincide with the central derivative of a weight 32 Siegelgenus 2 incoherent Eisenstein series. We will not include a further discussionof such higher codimension cases here.

In [Bru02], the first author generalized Borcherd’s regularized theta liftby allowing harmonic Maass forms as inputs. This provides an alternativeconstruction of Green functions for special divisors on Shimura varieties oforthogonal type, and has the advantage that one can try to use the method ofBorcherds to establish modularity of the generating series built with theseGreen functions. The main issue in doing this is that the divisor of the

MODULARITY OF UNITARY GENERATING SERIES 3

Borcherds forms is, a priori, only known on the generic fiber and hencemore detailed information is needed on its extension to the integral model.In the case of Hilbert modular surfaces, where the arithmetic cycles areintegral extensions of the Hirzebruch-Zagier curves T pmq, this is carried outin [BBGK07]. Recent advances in our knowledge of the integral modelsfor Hodge type Shimura varieties, [Madb, Mada], in particular the Shimuravarieties of orthogonal type over Q, suggest that a general result aboutmodularity of the generating series for classes of special divisors in arithmeticChow groups is now accessible.

In this paper we deal with unitary Shimura varieties for signature pn ´1, 1q. In this case a definition of arithmetic special cycles was given in [KR11,KR14], and was extended to the toroidal compactification in [How15]. TheBruinier-Borcherds construction of Green functions was carried over to theunitary case in [Hof14, BHY15]. The subject matter of the present paperis a proof of the modularity of the generating series for these classes ofarithmetic special divisors.

1.2. Statement of the main result. Fix a quadratic imaginary fieldk Ă C of odd discriminant discpkq “ ´D. We are concerned with thearithmetic of certain unitary Shimura varieties, whose definition depends onthe following initial data: let W0 and W be k-hermitian spaces of signaturep1, 0q and pn´1, 1q, respectively, where n ě 3. We assume that these spaceseach admit an Ok-lattice that is self-dual with respect to the hermitian form.

Attached to this data is a reductive algebraic group

(1.2.1) G Ă GUpW0q ˆGUpW q

over Q, defined as the subgroup on which the unitary similitude charactersare equal, and a compact open subgroup K Ă GpAf q depending on theabove choice of self-dual lattices. As explained in §2, there is an associatedhermitian symmetric domain D, and a stack ShpG,Dq over k whose complexpoints are identified with the orbifold quotient

ShpG,DqpCq “ GpQqzD ˆGpAf qK.

This is the unitary Shimura variety of the title.This stack can be interpreted as a moduli space of pairs pA0, Aq in which

A0 is an elliptic curve with complex multiplication by Ok, and A is a princi-pally polarized abelian scheme of dimension n endowed with an Ok-action.The pair pA0, Aq is required to satisfy some additional conditions, whichneed not concern us in the introduction.

Using the moduli interpretation, one can construct an integral model ofShpG,Dq over Ok. In fact, following work of Pappas and Kramer, we explainin §2.3 that there are two natural integral models related by a morphismSKra Ñ SPap. Each integral model has a canonical toroidal compactifica-tion whose boundary is a disjoint union of Cartier divisors, each of which


is smooth over Ok, and the above morphism extends uniquely to the com-pactifications:

(1.2.2) S˚Kra Ñ S˚Pap.

Each compactified integral model has its own desirable and undesirableproperties. For example, S˚Kra is regular, while S˚Pap is not. On the other

hand, every vertical (i.e. supported in nonzero characteristic) Weil divisoron S˚Pap has nonempty intersection with the boundary, while S˚Kra has certain

exceptional divisors in characteristics p | D that do not meet the boundary.An essential part of our method is to pass back and forth between these

two models in order to exploit the best properties of each, but for the sakeof simplicity we will state our main results solely in terms of the regularmodel S˚Kra. Denote by

Ch1QpS˚Kraq – PicpS˚Kraq bZ Q

the Chow group of rational equivalence classes of divisors with Q coefficients.In §2 we define a distinguished line bundle ω on SKra, called the line

bundle of weight one modular forms, and a family of Cartier divisors ZKrapmqindexed by integers m ą 0. These special divisors were introduced in [KR11,KR14], and studied further in [BHY15, How12, How15]. For the purposes ofthe introduction, we note only that one should regard the divisors as arisingfrom embeddings of smaller unitary groups into G.

Each special divisor ZKrapmq can be extended to a divisor on the toroidalcompactification simply by taking its Zariski closure, denoted Z˚Krapmq. Thetotal special divisor is defined as

(1.2.3) ZtotKrapmq “ Z˚Krapmq ` BKrapmq P Ch1

QpS˚Kraq

where the boundary contribution is defined, as in (5.3.3), by

BKrapmq “m

n´ 2

ÿ

Φ

#tx P L0 : xx, xy “ mu ¨ S˚KrapΦq.

The notation here is the following: The sum is over the equivalence classesof proper cusp label representatives Φ as defined in §3.1. These index theconnected components S˚KrapΦq Ă BS˚Kra of the boundary1. Inside the sum,pL0, x¨, ¨yq is a hermitian Ok-module of signature pn ´ 2, 0q, which dependson Φ.

The line bundle of modular forms ω admits a canonical extension to thetoroidal compactification, denoted the same way. For the sake of notationaluniformity, we also define (1.2.3) for m “ 0 by setting

ZtotKrap0q “ ω

´1 P Ch1QpS˚Kraq.

The following result appears in the text as Theorem 7.1.5.

1After base change to C, each S˚KrapΦq decomposes into h connected components, whereh is the class number of k.


Theorem A. Let χk : pZDZqˆ Ñ t˘1u be the Dirichlet character deter-mined by kQ. The formal generating series

ÿ

mě0

ZtotKrapmq ¨ q

m P Ch1QpS˚Kraqrrqss

is modular of weight n, level Γ0pDq, and character χnk in the following sense:

for every Q-linear functional ρ : Ch1QpS˚Kraq Ñ C, the series

ÿ

mě0

ρpZtotKrapmqq ¨ q

m P Crrqss

is the q-expansion of a classical modular form of the indicated weight, level,and character.

In fact we prove a stronger version of this theorem. Denote by xCh1

QpS˚Kraq

the Gillet-Soule [GS90] arithmetic Chow group of rational equivalence classes

of pairs pZ “ pZ,Grq, where Z is a divisor on S˚Kra with rational coefficients,and Gr is a Green function for Z. We allow the Green function to haveadditional log-log singularities along the boundary, as in the more generaltheory developed in [BGKK07]. See also [BBGK07, How15].

In §7.3 we use the theory of regularized theta lifts to construct Greenfunctions for the special divisors Ztot

Krapmq, and hence obtain arithmetic di-visors

pZtotKrapmq P

xCh1

QpS˚Kraq.

We also endow the line bundle ω with a metric that is smooth up to a log-logsingularity along the boundary. The resulting metrized line bundle pω thendefines a class

pZtotKrap0q “ pω´1 P xCh

1

QpS˚Kraq.

The following result appears in the text as Theorem 7.3.1.

Theorem B. The formal generating series

pφpτq “ÿ

mě0

pZtotKrapmq ¨ q

m P xCh1

QpS˚Kraqrrqss

is modular of weight n, level Γ0pDq, and character χnk, where modularity isunderstood in the same sense as Theorem A.

Remark 1.2.1. Theorem B implies that the Q-span of the classes pZtotKrapmq

is finite dimensional. See Remark 7.1.2.

Remark 1.2.2. There is a second method of constructing Green functions forthe special divisors, based on the methods of [Kud97b], which gives rise to a

non-holomorphic variant of pφpτq. It is a recent theorem of Ehlen-Sankaran[ES16] that Theorem B implies the modularity of this non-holomorphic gen-erating series. See §7.4.


The motivating desire for the modularity result of Theorem B is that itallows one to construct arithmetic theta lifts: If fpτq P SnpΓ0pDq, χ

nkq is a

classical scalar valued cusp form, we may form the Petersson inner product

pθpfqdef“ xfpτq, pφpτqyPet P

xCh1

CpS˚Kraq

as in [Kud04]. One expects, as in [loc. cit.], that the arithmetic intersecting

pairing of pθpfq against other cycle classes should be related to derivativesof L-functions, providing generalizations of the Gross-Zagier and Gross-Kohnen-Zagier theorems. Specific instances in which this expectation isfulfilled can be deduced from [BHY15, How12, How15]. This will be ex-plained in the companion paper [BHK`].

As this paper is rather long, we explain in the next two subsections themain ideas that go into the proof of Theorem A. The proof of Theorem Bis exactly the same, but one must keep track of Green functions.

1.3. Sketch of the proof, part I: the generic fiber. In this subsectionwe sketch the proof of modularity only in the generic fiber. That is, themodularity of

(1.3.1)ÿ

mě0

ZtotKrapmqk ¨ q

m P Ch1QpS˚Krakqrrqss.

As in [Bor99], the key to the proof is the study of Borcherds products [Bor98].A Borcherds product is a meromorphic modular form on an orthogonal

Shimura variety, whose construction depends on a choice of weakly holomor-phic input form, typically of negative weight. In our case the input form isany

(1.3.2) fpτq “ÿ

m"´8

cpmqqm PM !,82´npD,χ

n´2k q,

where the superscripts ! and 8 indicate that the weakly holomorphic formfpτq of weight 2 ´ n is allowed to have a pole at the cusp 8, but must beholomorphic at all other cusps. We assume also that all cpmq P Z.

Our Shimura variety ShpG,Dq admits a natural map to an orthogonalShimura variety. Indeed, the k-vector space

V “ HomkpW0,W q

admits a natural hermitian form x¨, ¨y of signature pn´ 1, 1q, induced by thehermitian forms on W0 and W . The natural action of G on V determinesan exact sequence

(1.3.3) 1 Ñ ReskQGm Ñ GÑ UpV q Ñ 1

of reductive groups over Q. If we now view V as a Q-vector space withquadratic form Qpxq “ xx, xy, we obtain a homomorphism G Ñ SOpV q.This homomorphism induces a map from ShpG,Dq to the Shimura varietyassociated with the group SOpV q. Note that the rational quadratic spacepV,Qq has signature p2n´ 2, 2q.


After possibly replacing f by a nonzero integer multiple, Borcherds con-structs a meromorphic modular form on the orthogonal Shimura varietydetermined by SOpV q, which can be pulled back to a meromorphic modu-lar form on ShpG,DqpCq. The result is a meromorphic section ψpfq of ωk,where the weight

k “ÿ

r|D

γr ¨ crp0q P Z

is the integer defined in §5.3. The constant γr “ś

p|r γp is a 4th root of

unity (with γ1 “ 1) and crp0q is the constant term of f at the cusp

8r “r

DP Γ0pDqzP1pQq,

in the sense of Definition 4.1.1.Initially, ψpfq is characterized by specifying ´ log ||ψpfq||, where || ¨ || is

the Petersson norm on ωk. In particular, ψpfq is only defined up to rescal-ing by a complex number of absolute value 1 on each connected componentof ShpG,DqpCq. We prove that, after a suitable rescaling, ψpfq is the an-alytification of a rational section of the line bundle ωk on ShpG,Dq. Inother words, the Borcherds product is algebraic and defined over the reflexfield k. This result is not really new, as one could first prove the analogousresult on the orthogonal Shimura variety using the explicit q-expansion ofBorcherds, from which the result on ShpG,Dq follows immediately. In anycase, the algebraicity and descent to the reflex field allow us to view ψpfq asa rational section of ωk both on the integral model SKra, and on its toroidalcompactification.

We compute the divisor of ψpfq on the generic fiber of the toroidal com-pactification S˚Krak, and find

(1.3.4) divpψpfqqk “ÿ

mą0

cp´mq ¨ ZtotKrapmqk.

The calculation of the divisor on the interior SKrak follows immediatelyfrom the corresponding calculations of Borcherds on the orthogonal Shimuravariety. The multiplicities of the boundary components are computed usingthe results of [Kud16], which describe the structure of the Fourier-Jacobiexpansion of ψpfq along each boundary component.

The cusp 81 “ 1D is Γ0pDq-equivalent to the usual cusp at 8, and soc1p0q “ cp0q. It follows from this and (1.3.4) that

(1.3.5) 0 “ÿ

mě0

cp´mq ¨ ZtotKrapmqk ´

ÿ

r|Drą1

γrcrp0q ¨ ω

in Ch1QpS˚Krakq. In §4.2 we construct for each r | D an Eisenstein series

Erpτq “ÿ

mě0

erpmq ¨ qm PMnpD,χ

nkq,


which, by a simple residue calculation, satisfies

crp0q “ ´ÿ

mą0

cp´mqerpmq.

Substituting this expression into (1.3.5) yields

(1.3.6) 0 “ÿ

mě0

cp´mq ¨´

ZtotKrapmqk `

ÿ

r|Drą1

γrerpmq ¨ ω¯

,

where we have also used the relation erp0q “ 0 for r ą 1.We now invoke a variant of the modularity criterion of [Bor99], which is

our Theorem 4.2.3: if a formal q-expansionÿ

mě0

dpmqqm P Crrqss

satisfies 0 “ř

mě0 cp´mqdpmq for every input form (1.3.2), then it must bethe q-expansion of a modular form of weight n, level Γ0pDq, and characterχnk. It follows immediately from this and (1.3.6) that the formal q-expansion

ÿ

mě0

´

ZtotKrapmqk `

ÿ

r|Drą1

γrerpmq ¨ ω¯

¨ qm

is modular in the sense of Theorem A. Rewriting this asÿ

mě0

ZtotKrapmqk ¨ q

m `ÿ

r|Drą1

γrErpτq ¨ ω

and using the modularity of each Eisenstein series Erpτq, we deduce that(1.3.1) is modular.

1.4. Sketch of the proof, part II: vertical components. In order toextend the arguments of §1.3 to prove Theorem A, it is clear that one shouldattempt to compute the divisor of ψpfq on the integral model S˚Kra and hopefor an expression similar to (1.3.4). Indeed, the bulk of this paper is devotedto precisely this problem.

The subtlety is that both divpψpfqq and ZtotKrapmq will turn out to have

vertical components supported in characteristics dividing D. Even worse, inthese bad characteristics there are exceptional divisors on S˚Kra that do notintersect the boundary, and the multiplicities of these divisors in divpψpfqqcannot be detected by examining the Fourier-Jacobi expansions of ψpfq.

This is where the second integral model S˚Pap plays an essential role. The

morphism (1.2.2) sits in a cartesian diagram

Exc //

S˚Kra

Sing // S˚Pap,


where the singular locus Sing ãÑ S˚Pap is the reduced closed substack of

points at which the structure morphism S˚Pap Ñ SpecpOkq is not smooth.It is 0-dimensional and supported in characteristics dividing D. The rightvertical arrow restricts to an isomorphism

(1.4.1) S˚Kra r Exc – S˚Pap r Sing.

For each connected component s P π0pSingq the fiber

Excs “ ExcˆS˚Paps

is a smooth, irreducible, vertical Cartier divisor on S˚Kra that does not meetthe boundary, and Exc “

Ů

s Excs. These are the exceptional divisors al-luded to above.

Using the fact that S˚Pap has geometrically normal fibers, one can see thatevery vertical divisor on it meets the boundary. Thus one could hope to use(1.4.1) to view ψpfq as a rational section on S˚Pap, compute its divisor thereby examining Fourier-Jacobi expansions, and then pull that calculation backto S˚Kra.

This is precisely what we do, but there is an added complication: Theline bundle ω on (1.4.1) does not extend to S˚Pap, and similarly the divisor

Z˚Krapmq on (1.4.1) cannot be extended across the singular locus to a Cartierdivisor on S˚Pap. However, if you square the line bundle and the divisors,they do extend across the singular locus: on S˚Pap, there is a unique line

bundle ΩPap whose restriction to (1.4.1) is isomorphic to ω2, and a uniqueCartier divisor Ytot

Pappmq whose restriction to (1.4.1) is equal to 2ZtotKrapmq.

Setting YtotPapp0q “ Ω´1

Pap, we obtain a formal generating series

ÿ

mě0

YtotPappmq ¨ q

m P Ch1QpS˚Papqrrqss

whose pullback via S˚Kra Ñ S˚Pap is twice the generating series of TheoremA, up to an error term coming from the exceptional divisors. More precisely,we show that the pullback is

2ÿ

mě0

ZtotKrapmq ¨ q

m ´ÿ

sPπ0pSingq

ϑspτq ¨ Excs P Ch1QpS˚Kraqrrqss,

where each ϑspτq is a classical theta function whose coefficients count pointsin a positive definite hermitian lattice Ls depending on s.

Over (1.4.1) we have ω2k – ΩkPap, which allows us to view ψpfq2 as a

rational section of the line bundle ΩkPap on S˚Pap. We examine its Fourier-

Jacobi expansions along the boundary components and are able to computeits divisor completely (it happens to include nontrivial vertical components).We then pull this calculation back to S˚Kra, and find that ψpfq, when viewed


as a rational section of ωk, has divisor

divpψpfqq “ÿ

mą0

cp´mq ¨ ZtotKrapmq ´

k

2¨ Exc`

ÿ

r|D

γrcrp0qÿ

p|r

S˚KraFp

´ÿ

mą0

cp´mq

2

ÿ

sPπ0pSingq

#tx P Ls : xx, xy “ mu ¨ Excs

where x¨, ÿ is the hermitian form on Ls, p Ă Ok is the unique prime abovep | D, and S˚KraFp

is the mod p fiber of S˚Kra, viewed as a divisor. This is

stated in the text as Theorem 5.3.3. Passing to the generic fiber recovers(1.3.4), as it must.

As in the argument leading to (1.3.6), this implies the relation

0 “ÿ

mě0

cp´mq ¨

¨

˝ZtotKrapmq ´

ÿ

sPπ0pSingq

#tx P Ls : xx, xy “ mu

2¨ Excs

˛

‚

`ÿ

mě0

cp´mq ¨ÿ

r|Drą1

γrerpmq

¨

˝ω Èxc

2´ÿ

p|r

S˚KraFp

˛

‚

in the Chow group of S˚Kra, and the modularity criterion implies thatÿ

mě0

ZtotKrapmq ¨ q

m ´1

2

ÿ

sPπ0pSingq

ϑspτq ¨ Excs

`ÿ

r|Drą1

γrErpτq ¨

¨

˝ω Èxc

2´ÿ

p|r

S˚KraFp

˛

‚

is a modular form. As the theta series ϑspτq and Eisenstein series Erpτq areall modular, so is

ř

ZtotKrapmqq

m. This completes the outline of the proof ofTheorem A.

1.5. The structure of the paper. We now briefly describe the contentsof the various sections of the paper.

In §2 we introduce the unitary Shimura variety associated to the groupG of (1.2.1), and explain its realization as a moduli space of pairs pA0, Aq ofabelian varieties with extra structure. We then review the integral modelsconstructed by Pappas and Kramer, and the singular and exceptional lociof these models. These are related by a diagram

Exc //

SKra

Sing // SPap,

where the vertical arrow on the right is an isomorphism outside of the 0-dimensional singular locus Sing. We also define the line bundle of modular


forms ω on SKra. The first main result of §2 is Theorem 2.4.2, which assertsthe existence of a bundle ΩPap on SPap restricting to ω2 over the complementof the singular locus. We then define the Cartier divisor ZKrapmq on SKra.The second main result of the section is Theorem 2.5.2, which asserts theexistence of a unique Cartier divisor YPappmq on SPap whose restriction tothe complement of the singular locus coincides with 2ZKrapmq. Finally,writing ΩKra and YKrapmq for the pullbacks of ΩPap and YPappmq to SKra,the third main result of the section, Theorem 2.6.3 in the text, gives thefollowing precise relations among these objects on the two models.

Theorem C. There is an isomorphism

ω2 bOpExcq – ΩKra

of line bundles on SKra, as well as an equality

2ZKrapmq “ YKrapmq `ÿ

sPπ0pSingq


of Cartier divisors. Here Ls is a positive definite self-dual hermitian latticeof rank n associated to the singular point s.

In §3 we describe the canonical toroidal compactifications S˚Kra Ñ S˚Pap,and the structure of their formal completions along the boundary. In §3.1and §3.2 we introduce the cusp labels Φ that index the cusps, and theirassociated mixed Shimura varieties. In §3.3 we construct smooth integralmodels CΦ of these mixed Shimura varieties, following the general recipesof the theory of arithmetic toroidal compactification, as moduli space of 1-motives. In §3.4 we give a second moduli interpretation of these integralmodels. The expressions for the Fourier-Jacobi expansions based on thissecond interpretation are more easily related to Fourier-Jacobi expansionson orthogonal Shimura varieties. This is one of the key technical steps inour work; see the remarks at the beginning of §3 for further discussion. In§3.5 and §3.6 we describe the line bundle of modular forms and the spe-cial divisors on the boundary charts. Theorem 3.7.1 describes the canonicaltoroidal compactifications S˚Kra and S˚Pap and their properties. In §3.8 we

describe the Fourier-Jacobi expansions of sections of ωk on S˚Kra in alge-braic language, and in §3.9 we explain how to express these Fourier-Jacobicoefficients in classical complex analytic coordinates.

In §4 we review the weakly holomorphic modular forms f P M !,8k pD,χq,

whose regularized theta lifts are used to define the Borcherds forms thatultimately provide relations in (arithmetic) Chow groups. We also state theversion of Borcherd modularity criterion, Theorem 4.2.3, which we need.

In §5 we consider unitary Borcherds forms associated to weakly holo-

morphic forms f P M !,82´npD,χ

n´2k q. Ultimately, the integrality properties

of the unitary Borcherds forms will be deduced from an analysis of theirFourier-Jacobi expansions. These expansions involve certain products ofJacobi theta functions, and so, in §5 we review facts about the arithmetic


theory of Jacobi forms, viewed as sections of a suitable line bundle Jk,mon the universal elliptic curve E Ñ Y over Z. The key point is to havea precise description of the divisor of the basic section Θ24 P H0pE ,J0,12q

of Proposition 5.1.4. In §5.2 we prove Borcherds quadratic identity, allow-ing us to relate J0,1 to a certain line bundle determined by a Borcherdsform on the boundary component BΦ associated to a cusp label Φ. Afterthese technical preliminaries, we come to the statements of our main re-sults about unitary Borcherds forms. Theorem 5.3.1 asserts that, for each

f P M !,82´npD,χ

n´2k q satisfying integrality conditions on the Fourier coeffi-

cients cp´mq with m ě 0, there is a rational section ψpfq of the line bundleωk on S˚Kra with explicit divisor on the generic fiber and prescribed zerosand poles along each boundary component. Moreover, for each cusp labelΦ, the leading Fourier-Jacobi coefficient ψ0 of ψpfq has an expression as aproduct of three factors, two of which, P vertΦ and P horΦ , are constructed interms of Θ24. Theorem 5.3.3 gives the precise divisor of ψpfq on S˚Kra, andTheorem 5.3.4 gives an analogous formula on S˚Pap. In §5.4 we compute the

divisors on BΦ of the factors P vertΦ and P horΦ .In §6 we prove the main results stated in §5. In §6.1 we construct a

vector valued form f from f P M !,82´npD,χ

n´2k q, and give expressions for

its Fourier coefficients in terms of those of f . The vector valued form fdefines a Borcherds form ψpfq on the symmetric space D for the orthogo-nal group of the quadratic space pV,Qq and, in §6.2, we obtain the unitaryBorcherds form ψpfq as its pullback to D. In §6.3 we determine the analyticFourier-Jacobi expansion of ψpfq at the cusp Φ by pulling back the prod-

uct formula for ψpfq along a one-dimensional boundary component of D.In §6.4 we show that the unitary Borcherds form constructed analyticallyarises from a rational section of ωk and that, after rescaling, this section isdefined over k. This is Proposition 6.4.3. In §6.5 we complete the proofs ofTheorems 5.3.1, 5.3.3, and 5.3.4.

In §7 we give the proof of the modularity results discussed in detail earlierin the introduction.

Finally, in §8, we provide some technical details omitted from or supple-mentary to the earlier sections.

1.6. Thanks. The results of this paper are the outcome of a long termproject, begun initially in Bonn in June of 2013, and supported in a crucialway by three weeklong meetings at AIM, in Palo Alto (May of 2014) and SanJose (November of 2015 and 2016), as part of their AIM SQuaRE’s program.The opportunity to spend these periods of intensely focused efforts on theproblems involved was essential. We would like to thank the University ofBonn and AIM for their support.

1.7. Notation. Throughout the paper, k Ă C is a quadratic imaginary fieldof odd discriminant discpkq “ ´D. Denote by δ “

?´D P k the unique

choice of square root with Impδq ą 0. Let kab Ă C be the maximal abelian


extension of k in C, and let

art : kˆzpkˆ Ñ Galpkabkq

be the Artin map of class field theory, normalized as in [Mil05, §11]. Asusual, S “ ResCRGm is Deligne’s torus.

Fix a π P Ok satisfying Ok “ Z` Zπ. If S is any Ok-scheme, define

εS “ π b 1´ 1b iSpπq P Ok bZ OS

εS “ π b 1´ 1b iSpπq P Ok bZ OS ,

where iS : Ok Ñ OS is the structure map. The ideals generated by theseelements are independent of the choice of π, and sit in exact sequences offree OS-modules

0 Ñ pεSq Ñ Ok bZ OSαbx ÞÑiSpαqxÝÝÝÝÝÝÝÝÑ OS Ñ 0

and

0 Ñ pεSq Ñ Ok bZ OSαbx ÞÑiSpαqxÝÝÝÝÝÝÝÝÑ OS Ñ 0.

If N is an OkbZOS-module then NεSN is the maximal quotient of N onwhich Ok acts through the structure morphism iS : Ok Ñ OS , and NεSNis the maximal quotient on which Ok acts through the conjugate of thestructure morphism. If D P OˆS then more is true: there is a decomposition

(1.7.1) N “ εSN ‘ εSN,

and the summands are the maximal submodules on which Ok acts throughthe structure morphism and its conjugate, respectively. From this discussionit is clear that one should regard εS and εS as integral substitutes for theorthogonal idempotents in k bQ C – Cˆ C. The Ok-scheme S will usuallybe clear from context, and we abbreviate εS and εS to ε and ε.

For a prime p ď 8 we write pa, bqp for the Hilbert symbol of a, b P Qˆp .Recall that the invariant of a hermitian space V over kp “ k bQ Qp isdefined by

invppV q “ pdetV,´Dqp,(1.7.2)

where detV is the determinant of the matrix of the hermitian form withrespect to a kp-basis. If p ă 8 then V is determined up to isomorphism byits kp-rank and invariant. If p “ 8 then V is determined up to isomorphismby its signature pr, sq, and its invariant satisfies inv8pV q “ p´1qs.

2. Unitary Shimura varieties

In this section we define a unitary Shimura variety ShpG,Dq over k anddescribe its moduli interpretation. We then recall the work of Pappas andKramer, which provides us with two integral models related by a surjectionSKra Ñ SPap. This surjection becomes an isomorphism after restriction toOkr1Ds. We define a line bundle of weight one modular forms ω and afamily of Cartier divisors ZKrapmq, m ą 0, on SKra,


The line bundle ω and the divisors ZKrapmq do not descend to SPap, andthe main original material in §2 is the construction of suitable substituteson SPap. These substitutes consist of a line bundle ΩPap that agrees with ω2

after restricting to Okr1Ds, and Cartier divisors YPappmq that agree with2ZKrapmq after restricting to Okr1Ds.

2.1. The Shimura variety. Let W0 and W be k-vector spaces endowedwith hermitian forms H0 and H of signatures p1, 0q and pn ´ 1, 1q, respec-tively. We always assume that n ě 3. Abbreviate

W pRq “W bQ R, W pCq “W bQ C, W pAf q “W bQ Af ,

and similarly for W0. We assume the existence of Ok-lattices a0 Ă W0 anda Ă W , self-dual with respect to the hermitian forms H0 and H. This isequivalent to self-duality with respect to the symplectic forms

(2.1.1) ψ0pw,w1q “ TrkQH0pδ

´1w,w1q, ψpw,w1q “ TrkQHpδ´1w,w1q.

This data will remain fixed throughout the paper.As in (1.2.1), let G Ă GUpW0q ˆ GUpW q be the subgroup of pairs for

which the similitude factors are equal. The common similitude character isdenoted ν : GÑ Gm. The three reductive groups in (1.2.1) determine threehermitian symmetric domains: Let

DpW0q “ ty0u

be a one-point set, let

(2.1.2) DpW q “ tnegative definite k-stable R-planes y ĂW pRqu,

and set

D “ DpW0q ˆDpW q.The lattices a0 and a determine a maximal compact open subgroup

(2.1.3) K “

g P GpAf q : gpa0 Ă pa0 and gpa Ă pa(

Ă GpAf q,

and the orbifold quotient

ShpG,DqpCq “ GpQqzD ˆGpAf qKis the space of complex points of a smooth k-stack of dimension n ´ 1,denoted ShpG,Dq.

The symplectic forms (2.1.1) determine a k-conjugate-linear isomorphism

(2.1.4) HomkpW0,W qx ÞÑx_ÝÝÝÝÑ HomkpW,W0q,

characterized by ψpxw0, wq “ ψ0pw0, x_wq. The k-vector space

V “ HomkpW0,W q

carries a hermitian form of signature pn´ 1, 1q defined by

(2.1.5) xx1, x2y “ x_2 ˝ x1 P EndkpW0q – k.

The group G acts on V in a natural way, defining an exact sequence (1.3.3).


The hermitian form on V induces a quadratic form Qpxq “ xx, xy, withassociated Q-bilinear form

(2.1.6) rx, ys “ TrkQxx, yy.

In particular, we obtain a representation GÑ SOpV q.

Proposition 2.1.1. The stack ShpG,DqC has 21´opDqh2 connected compo-nents, where h is the class number of k and opDq is the number of primedivisors of D. Every component is defined over the Hilbert class field H of k,and the set of components is a disjoint union of simply transitive GalpHkq-orbits.

Proof. Each g P GpAf q determines Ok-lattices

ga0 “W0 X gpa0, ga “W X gpa.

The hermitian forms H0 and H need not be Ok-valued on these lattices.However, if ratpνpgqq denotes the unique positive rational number such that

νpgq

ratpνpgqqP pZˆ

then the rescaled hermitian forms ratpνpgqq´1H0 and ratpνpgqq´1H makega0 and ga into self-dual hermitian lattices.

As D is connected, the components of ShpG,DqC are in bijection with theset GpQqzGpAf qK. The function g ÞÑ pga0, gaq then establishes a bijectionfrom GpQqzGpAf qK to the set of isometry classes of pairs of self-dual her-mitian Ok-lattices pa10, a

1q of signatures p1, 0q and pn´1, 1q, respectively, forwhich the self-dual hermitian lattice HomOk

pa10, a1q lies in the same genus as

HomOkpa0, aq Ă V .

Using the fact that SUpV q satisfies strong approximation, one can show

that there are exactly 21´opDqh isometry classes in the genus of HomOkpa0, aq,

and each isometry class arises from exactly h isometry classes of pairs pa10, a1q.

Identifying the connected components with pairs of lattices as above,Deligne’s reciprocity law [Mil05, §13] reads as follows: if σ P AutpCkqand s P pkˆ Ă Gppkq are related by artpsq “ σ|kab , then

pa10, a1qσ “ psa10, sa

1q.

The right hand side obviously only depends on the image of s in kˆzpkˆ pOˆk ,and all claims about the Galois action follow easily.

It will be useful at times to have other interpretations of the hermitiandomain D. The following remarks provide alternate points of view.

Remark 2.1.2. Define a Hodge structure

Fil1W0pCq “ 0, Fil0W0pCq “ εW0pCq, Fil´1W0pCq “W0pCqon W0pCq, and identify the unique point y0 P DpW0q with the correspondingmorphism SÑ GUpW0qR. Every y P DpW q defines a Hodge structure

Fil1W pCq “ 0, Fil0W pCq “ εy ‘ εyK, Fil´1W pCq “W pCq


on W pCq, where εy and εyK denote the images of y and yK under

W pRq ÑW pCq εÝÑ εW pCq

W pRq ÑW pCq εÝÑ εW pCq.

If we identify y P DpW q with the corresponding morphism S Ñ GUpW qR,then for any point z “ py0, yq P D the product morphism

y0 ˆ y : SÑ GUpW0qR ˆGUpW qR

takes values in GR. This realizes D Ă HompS, GRq as a GpRq-conjugacyclass.

Remark 2.1.3. Each pair py0, yq P D determines a line εy Ă W pCq, andhence a line

w “ HomCpW0pCqεW0pCq, εyq Ă εV pCq.This construction identifies D –

w P εV pCq : rw,ws ă 0(

Cˆ.

Remark 2.1.4. In fact, the discussion above shows that ShpG,Dq admitsa map to the Shimura variety defined the group UpV q together with thehomomorphism

hGross : SÑ UpV qpRq, z ÞÑ diagp1, . . . , 1,z

zq.

Here we have chosen a basis for V pRq for which the hermitian form hasmatrix diagp1n´1,´1q. Note that, for analogous choices of bases for W0pRqand W pRq, the corresponding map is

h : SÑ GpRq, z ÞÑ pzq ˆ diagpz, . . . , z, zq,

which, under composition with the homomorphism GpRq Ñ UpV qpRq, giveshGross. The existence of this map provides an answer to a question posedby Gross: how can one explicitly relate the Shimura variety defined bythe unitary group UpV q (as opposed to the Shimura variety defined by thesimilitude group GUpV q) to a moduli space of abelian varieties? Our answeris that Gross’s unitary Shimura variety is a quotient of our space ShpG,Dq,whose interpretation as a moduli space is explained in the next section.

2.2. Moduli interpretation. We wish to interpret ShpG,Dq as a modulispace of pairs of abelian varieties with additional structure. First, we recallsome generalities on abelian schemes.

For an abelian scheme π : A Ñ S over an arbitrary base S, define thefirst relative de Rham cohomology sheaf H1

dRpAq “ R1π˚Ω‚AS as the relative

hypercohomology of the de Rham complex Ω‚AS . The relative de Rham

homology

HdR1 pAq “ HompH1

dRpAq,OSq

is a locally free OS-module of rank 2 ¨ dimpAq, sitting in an exact sequence

0 Ñ Fil0HdR1 pAq Ñ HdR

1 pAq Ñ LiepAq Ñ 0.


Any polarization of A induces an OS-valued alternating pairing on HdR1 pAq,

which in turn induces a pairing

(2.2.1) Fil0HdR1 pAq b LiepAq Ñ OS .

If the polarization is principal then both pairings are perfect. When S “SpecpCq, Betti homology satisfies H1pA,Cq – HdR

1 pAq, and

ApCq – H1pA,ZqzHdR1 pAqFil0HdR

1 pAq.

For any pair of nonnegative integers ps, tq, define an algebraic stack Mps,tq

over k as follows: for any k-scheme S let Mps,tqpSq be the groupoid of triplespA, ι, ψq in which

‚ AÑ S is an abelian scheme of relative dimension s` t,‚ ι : Ok Ñ EndpAq is an action such that the locally free summands

LiepAq “ εLiepAq ‘ εLiepAq

of (1.7.1) have OS-ranks s and t, respectively,‚ ψ : AÑ A_ is a principal polarization, such that the induced Rosati

involution : on End0pAq satisfies ιpαq: “ ιpαq for all α P Ok.

We usually omit ι and ψ from the notation, and just write A PMps,tqpSq.

Proposition 2.2.1. The Shimura variety ShpG,Dq is isomorphic to an openand closed substack

(2.2.2) ShpG,Dq ĂMp1,0q ˆk Mpn´1,1q.

More precisely, ShpG,DqpSq classifies, for any k-scheme S, pairs

(2.2.3) pA0, Aq PMp1,0qpSq ˆMpn´1,1qpSq

for which there exists, at every geometric point s Ñ S, an isomorphism ofhermitian Ok,`-modules

(2.2.4) HomOkpTÀ0,s, TÀsq – HomOk

pa0, aq b Z`

for every prime `. Here the hermitian form on the right hand side of (2.2.4)is the restriction of the hermitian form (2.1.5) on HomkpW0,W qbQ`. Thehermitian form on the left hand side is defined similarly, replacing the sym-plectic forms (2.1.1) on W0 and W with the Weil pairings on the Tate mod-ules TÀ0,s and TÀs.

Proof. As this is routine, we only describe the open and closed immersionon complex points. Fix a point

pz, gq P ShpG,DqpCq.

As in the proof of Proposition 2.1.1, g determines Ok-lattices ga0 Ă W0

and ga Ă W , which are self-dual with respect to the symplectic formsratpνpgqq´1ψ0 and ratpνpgqq´1ψ obtained by rescaling (2.1.1).


By Remark 2.1.2 the point z P D determines Hodge structures on W0 andW , and in this way pz, gq determines principally polarized complex abelianvarieties

A0pCq “ ga0zW0pCqFil0pW0q

ApCq “ gazW pCqFil0pW q

with actions of Ok. One can easily check that the pair pA0, Aq determinesa complex point of Mp1,0q ˆk Mpn´1,1q, and this construction defines (2.2.2)on complex points.

The following lemma will be needed in §2.3 for the construction of integralmodels for ShpG,Dq.

Lemma 2.2.2. Fix a k-scheme S, a geometric point sÑ S, a prime p, anda point (2.2.3). If the relation (2.2.4) holds for all ` ‰ p, then it also holdsfor ` “ p.

Proof. As the stack ShpG,Dq is of finite type over k, we may assume thats “ SpecpCq. The polarizations on A0 and A induce symplectic formson the first homology groups H1pA0,spCq,Zq and H1pAspCq,Zq, and theconstruction (2.1.5) makes

LBepA0,s, Asq “ HomOk

`

H1pA0,spCq,Zq, H1pAspCq,Zq˘

into a self-dual hermitian Ok-lattice of signature pn´ 1, 1q, satisfying

LBepA0,s, Asq bZ Z` – HomOkpTÀ0,s, TÀsq

for all primes `.If the relation (2.2.4) holds for all primes ` ‰ p, then LBepA0,s, Asq b Q

and HomkpW0,W q are isomorphic as k-hermitian spaces everywhere locallyexcept at p, and so they are isomorphic at p as well. In particular, for every` (including ` “ p) both sides of (2.2.4) are isomorphic to self-dual latticesin the hermitian space HomkpW0,W q bQ Q`. By the results of Jacobowitz[Jac62] all self-dual lattices in this local hermitian space are isomorphic2,and so (2.2.4) holds for all `.

Remark 2.2.3. For any positive integer m define

Kpmq “ ker`

K Ñ AutOkppa0mpa0q ÂutOk

ppampaq˘

.

For a k-scheme S, a Kpmq-structure on pA0, Aq P ShpG,DqpSq is a triplepα0, α, ζq in which ζ : µm – ZmZ is an isomorphism of S-group schemes,and

α0 : A0rms – pa0mpa0, α : Arms – pampa

are Ok-linear isomorphisms identifying the Weil pairings on A0rms andArms with the ZmZ-valued symplectic forms on pa0mpa0 and pampa deduced

2This uses our standing hypothesis that D is odd.


from the pairings (2.1.1). The Shimura variety GpQqzDˆGpAf qKpmq ad-mits a canonical model over k, parametrizing Kpmq-structures on points ofShpG,Dq.

2.3. Integral models. In this subsection we describe two integral modelsof ShpG,Dq over Ok, related by a morphism SKra Ñ SPap.

The first step is to construct an integral model of the moduli space Mp1,0q.More generally, we will construct an integral model of Mps,0q for any s ą 0.Define an Ok-stack Mps,0q as the moduli space of triples pA, ι, ψq over Ok-schemes S such that

‚ AÑ S is an abelian scheme of relative dimension s,‚ ι : Ok Ñ EndpAq is an action such εLiepAq “ 0, or, equivalently,

such that that the induced action of Ok on the OS-module LiepAqis through the structure map iS : Ok Ñ OS ,

‚ ψ : A Ñ A_ is a principal polarization whose Rosati involutionsatisfies ιpαq: “ ιpαq for all α P Ok.

The stack Mps,0q is smooth of relative dimension 0 over Ok by [How15,Proposition 2.1.2], and its generic fiber is the stack Mps,0q defined earlier.

The question of integral models for Mpn´1,1q is more subtle, but well-understood after work of Pappas and Kramer. The first integral model wasdefined by Pappas: let

MPappn´1,1q Ñ SpecpOkq

be the algebraic stack whose functor of points assigns to an Ok-scheme Sthe groupoid of triples pA, ι, ψq in which

‚ AÑ S is an abelian scheme of relative dimension n,‚ ι : Ok Ñ EndpAq is an action satisfying the determinant condition

detpT ´ ιpαq | LiepAqq “ pT ´ αqn´1pT ´ αq P OSrT s

for all α P Ok,‚ ψ : A Ñ A_ is a principal polarization whose Rosati involution

satisfies ιpαq: “ ιpαq for all α P Ok,‚ viewing the elements εS and εS of §1.7 as endomorphisms of LiepAq,

the induced endomorphismsľn

εS :ľn

LiepAq Ñľn

LiepAqľ2

εS :ľ2

LiepAq Ñľ2

LiepAq

are trivial (Pappas’s wedge condition).

It is clear that the generic fiber of MPappn´1,1q is isomorphic to the moduli

space Mpn´1,1q defined earlier. Denote by

Singpn´1,1q ĂMPappn´1,1q

the singular locus: the reduced substack of points at which the structuremorphism to Ok is not smooth.


Theorem 2.3.1 (Pappas [Pap00]). The stack MPappn´1,1q is flat over Ok of

relative dimension n´ 1, and is Cohen-Macaulay and normal. Moreover:

(1) For any prime p Ă Ok, the reduction MPappn´1,1qFp

is Cohen-Macaulay

and geometrically normal.(2) The singular locus is a 0-dimensional stack, finite over Ok and sup-

ported in characteristics dividing D. It is the reduced substack un-derlying the closed substack defined by δ ¨ LiepAq “ 0.

(3) The stack MPappn´1,1q becomes regular after blow-up along the singular

locus.

The blow up of MPappn´1,1q along the singular locus is denoted

(2.3.1) MKrapn´1,1q ÑMPap

pn´1,1q.

Kramer has shown that its functor of points assigns to an Ok-scheme S thegroupoid of quadruples pA, ι, ψ,FAq in which

‚ AÑ S is an abelian scheme of relative dimension n,‚ ι : Ok Ñ EndpAq is an action of Ok,‚ ψ : AÑ A_ is a principal polarization satisfying ιpαq: “ ιpαq for allα P Ok,

‚ FA Ă LiepAq is an Ok-stable OS-module local direct summand ofrank n ´ 1 satisfying Kramer’s condition: Ok acts on FA via thestructure map Ok Ñ OS , and acts on the line bundle LiepAqFA viathe complex conjugate of the structure map.

The morphism (2.3.1) simply forgets the subsheaf FA.Recalling (2.2.2), we define our first integral model

SPap ĂMp1,0q ˆMPappn´1,1q

as the Zariski closure of ShpG,Dq in the fiber product on the right, which,like all fiber products below, is taken over over SpecpOkq. Using Lemma2.2.2, one can show that it is characterized asthe open and closed substackwhose functor of points assigns to any Ok-scheme S the groupoid of pairs

pA0, Aq PMp1,0qpSq ˆMPappn´1,1qpSq

such that, at any geometric point s Ñ S, the relation (2.2.4) holds for allprimes ` ‰ charpkpsqq.

Our second integral model of ShpG,Dq is defined as the cartesian product

SKra//

Mp1,0q ˆMKrapn´1,1q

SPap//Mp1,0q ˆMPap

pn´1,1q.


The singular locus Sing Ă SPap and exceptional locus Exc Ă SKra are definedby the cartesian squares

Exc //

SKra

Sing //

SPap

Mp1,0q ˆ Singpn´1,1q//Mp1,0q ˆMPap

pn´1,1q.

Both are proper over Ok, and supported in characteristics dividing D.

Theorem 2.3.2 (Kramer [Kra03], Pappas [Pap00]). The Ok-stack SKra isregular and flat. The Ok-stack SPap is Cohen-Macaulay and normal, withCohen-Macaulay and geometrically normal fibers. Furthermore:

(1) The exceptional locus Exc Ă SKra is a disjoint union of smoothCartier divisors. The singular locus Sing Ă SPap is a reduced closedstack of dimension 0, supported in characteristics dividing D.

(2) The fiber of Exc over a geometric point s Ñ Sing is isomorphic tothe projective space Pn´1 over kpsq.

(3) The morphism SKra Ñ SPap is surjective, and restricts to an iso-morphism

(2.3.2) SKra r Exc – SPap r Sing.

For an Ok-scheme S, the inverse of this isomorphism endows

pA0, Aq P`

SPap r Sing˘

pSq

with the subsheaf FA “ ker`

ε : LiepAq Ñ LiepAq˘

.

2.4. The line bundle of modular forms. By Remark 2.1.2, every pointz P D determines Hodge structures on W0 and W of weight ´1, and hencea Hodge structure of weight 0 on V “ HomkpW0,W q.

Consider the holomorphic line bundle ωan on D whose fiber at z “py0, yq P D is the complex line ωanz “ Fil1V pCq. Somewhat more concretely,

(2.4.1) ωanz “ HomCpW0pCqεW0pCq, εyq Ă V pCq.

If we embed D into projective space over εV pCq as in Remark 2.1.3, this issimply the restriction of the tautological bundle. There is an obvious actionof GpRq on the total space of ωan, lifting the natural action on D, and soωan descends to a line bundle on the complex orbifold ShpG,DqpCq.

The holomorphic line bundle just define is algebraic, has a canonical modelover the reflex field, and extends in a natural way to the integral model SKra,as the following proposition demonstrates.


Proposition 2.4.1. Let pA0, Aq be the universal object over SKra, and letFA Ă LiepAq be the universal subsheaf of Kramer’s moduli problem. Theline bundle ω on SKra defined by

ω´1 “ LiepA0q b LiepAqFArestricts to the already defined ωan in the complex fiber.

Proof. Working in the complex fiber of SKra, Theorem 2.3.2 implies

εLiepAq “ ker`

ε : LiepAq Ñ LiepAq˘

“ FA,and so ω´1 – LiepA0q b LiepAqεLiepAq. The perfect pairing (2.2.1) nowinduces an isomorphism

ω – Hom`

LiepA0q, εFil0HdR1 pAq

˘

.

By examining the proof of Proposition 2.2.1, this agrees with ωan.

We call ω the line bundle of weight one modular forms on SKra. It doesnot descend to SPap, but it is closely related to another line bundle thatdoes. This is the content of the following theorem, whose proof will occupythe remainder of §2.4.

Theorem 2.4.2. There is a unique line bundle ΩPap on SPap whose re-striction to the nonsingular locus (2.3.2) is isomorphic to ω2. We denoteby ΩKra its pullback via SKra Ñ SPap.

Let pA0, Aq be the universal object over SPap, and recall the short exactsequence

0 Ñ Fil0HdR1 pAq Ñ HdR

1 pAqqÝÑ LiepAq Ñ 0

of vector bundles on SPap. The subsheaf Fil0HdR1 pAq is maximal isotropic

with respect to the perfect alternating pairing

ψ : HdR1 pAq bHdR

1 pAq Ñ OSPap

induced by the principal polarization on A. As HdR1 pAq is a locally free

Ok bZ OSPap-module of rank n, the quotient HdR

1 pAqεHdR1 pAq is a rank n

vector bundle.The vector bundle HdR

1 pA0q is locally free of rank one over Ok bZ OSPap

and, by definition of the moduli problem defining SPap, its quotient LiepA0q

is annihilated by ε. From this it is not hard to see that

(2.4.2) Fil0HdR1 pA0q “ εHdR

1 pA0q.

Define a line bundle

PPap “ Hom´

ľnHdR

1 pAqεHdR1 pAq,

ľnLiepAq

¯

on SPap, and denote by PKra its pullback via SKra Ñ SPap. If a and b are

local sections of HdR1 pAq, define a local section Pabb of PPap by

Pabbpe1 ^ ¨ ¨ ¨ ^ enq “nÿ

k“1

p´1qk`1 ¨ ψpεa, ekq ¨ qpεbq ^ qpe1q ^ ¨ ¨ ¨ ^ qpenqloooooooooomoooooooooon

omit qpekq

.


If we modify any ek by a section of εHdR1 pAq, the right hand side is un-

changed; this is a consequence of the vanishing ofľ2

ε :ľ2

LiepAq Ñľ2

LiepAq

imposed in the moduli problem defining SPap. We now have a morphism

(2.4.3) P : HdR1 pAq bHdR

1 pAq ÞÑ PPap

defined by ab b ÞÑ Pabb.

Proposition 2.4.3. The morphism P factors through a morphism

P : LiepAq b LiepAq Ñ PPap.

After pullback to SKra there is a further factorization

(2.4.4) P : LiepAqFA b LiepAqFA Ñ PKra,

and this map becomes an isomorphism after restriction to SKra r Exc .

Proof. Let a and b be local sections of HdR1 pAq.

Assume first that a is contained in Fil0HdR1 pAq. In this case Pabb factors

through a mapľn

LiepAqεLiepAq Ñľn

LiepAq.

In the generic fiber of SPap, the sheaf LiepAqεLiepAq is a vector bundle ofrank n´ 1. This proves that Pabb is trivial over the generic fiber. As Pabbis a morphism of vector bundles on a flat Ok-stack, we deduce Pabb “ 0identically on SPap.

If instead b is contained in Fil0HdR1 pAq then qpεbq “ 0, and again Pabb “ 0.

These calculations prove that P factors through LiepAq b LiepAq.Now pullback to SKra. We need to check that Pabb vanishes if either of

a or b lies in FA. Once again it suffices to check this in the generic fiber,where it is clear from

(2.4.5) FA “ kerpε : LiepAq Ñ LiepAqq.

Over SKra we now have a factorization (2.4.4), and it only remains tocheck that its restriction to (2.3.2) is an isomorphism. For this, it sufficesto verify that (2.4.4) is surjective on the fiber at any geometric point

s “ SpecpFq Ñ SKra r Exc.

First suppose that charpFq is prime to D. In this case ε, ε P Ok bZ F areorthogonal idempotents, FAs “ εLiepAsq, and we may choose an Ok bZ F-basis e1, . . . , en P H

dR1 pAsq in such a way that

εe1, εe2, . . . , εen P Fil0HdR1 pAsq

andqpεe1q, qpεe2q, . . . , qpεenq P LiepAsq

are F-bases. This implies that

Pe1be1pe1 ^ ¨ ¨ ¨ ^ enq “ ψpεe1, εe1q ¨ qpεe1q ^ qpεe2q ^ ¨ ¨ ¨ ^ qpεenq ‰ 0,


and so

Pe1be1 P Hom`

ľnHdR

1 pAsqεHdR1 pAsq,

ľnLiepAsq

˘

is a generator. Thus P is surjective in the fiber at z.Now suppose that charpFq divides D. In this case there is an isomorphism

Frxspx2qx ÞÑε“εÝÝÝÝÑ Ok bZ F.

By Theorem 2.3.2 the relation (2.4.5) holds in an etale neighborhood of s,and it follows that we may choose an OkbZ F-basis e1, . . . , en P H

dR1 pAsq in

such a way thate2, εe2, εe3, . . . , εen P Fil0HdR

1 pAsq

andqpe1q, qpεe1q, qpe3q . . . , qpenq P LiepAsq

are F-bases. This implies that

Pe1be1pe1 ^ ¨ ¨ ¨ ^ enq “ ψpεe1, e2q ¨ qpεe1q ^ qpe1q ^ qpe3q ^ ¨ ¨ ¨ ^ qpenq ‰ 0,

and so, as above, P is surjective in the fiber at z.

Proof of Theorem 2.4.2. The uniqueness of ΩPap is clear: as Sing Ă SPap isa codimension ě 2 closed substack of a normal stack, any line bundle onthe complement of Sing admits at most one extension to all of SPap. Forexistence we define

ΩPap “ LiepA0qb2 b PPap,

and let ΩKra be its pullback via SKra Ñ SPap. Tensoring both sides of themorphism

pLiepAqFAqb2 Ñ PPap

of Proposition 2.4.3 with LiepA0qb2 defines a morphism ω2 Ñ ΩKra, whose

restriction to SKra r Exc is an isomorphism. In particular ω2 and ΩPap areisomorphic over (2.3.2).

2.5. Special divisors. Suppose S is a connected Ok-scheme, and

pA0, Aq P SPappSq.

Imitating the construction of (2.1.5), there is a positive definite hermitianform on HomOk

pA0, Aq defined by

(2.5.1) xx1, x2y “ x_2 ˝ x1 P EndOkpA0q – Ok,

whereHomOk

pA0, Aqx ÞÑx_ÝÝÝÝÑ HomOk

pA,A0q

is the Ok-conjugate-linear isomorphism induced by the principal polariza-tions on A0 and A.

For any positive m P Z, define the Ok-stack ZPappmq as the moduli stackassigning to a connected Ok-scheme S the groupoid of triples pA0, A, xq,where

‚ pA0, Aq P SPappSq,‚ x P HomOk

pA0, Aq satisfies xx, xy “ m.


Define a stack ZKrapmq in exactly the same way, but replacing SPap by SKra.Thus we obtain a cartesian diagram

ZKrapmq //

SKra

ZPappmq // SPap,

in which the horizontal arrows are relatively representable, finite, and un-ramified.

Each ZKrapmq is, etale locally on SKra, a disjoint union of Cartier divisors.More precisely, around any geometric point of SKra one can find an etaleneighborhood U with the property that the morphism ZKrapmqU Ñ U re-stricts to a closed immersion on every connected component Z Ă ZKrapmqU ,and Z Ă U is defined locally by one equation; this is [How15, Proposi-tion 3.2.3]. Summing over all connected components Z allows us to viewZKrapmqU as a Cartier divisor on U , and glueing as U varies over an etalecover defines a Cartier divisor on SKra, which we again denote by ZKrapmq.

Remark 2.5.1. It follows from (2.3.2) and the paragraph above that ZPappmqis locally defined by one equation away from the singular locus. However,this property may fail at points of Sing, and so ZPappmq does not define aCartier divisor on all of SPap.

The following theorem, whose proof will occupy the remainder of §2.5,shows that ZKrapmq is closely related to another Cartier divisor on SKra

that descends to SPap.

Theorem 2.5.2. For every m ą 0 there is a unique Cartier divisor YPappmqon SPap whose restriction to (2.3.2) is equal to the square of the Cartierdivisor ZKrapmq. We denote by YKrapmq its pullback via SKra Ñ SPap.

Remark 2.5.3. The Cartier divisor YPappmq is flat over Ok, as is the re-striction of ZKrapmq to SKra r Exc. This will be proved in Corollary 3.7.2,by studying the structure of the divisors near the boundary of a toroidalcompactification.

The map ZPappmq Ñ SPap is finite, unramified, and relatively repre-sentable. It follows that every geometric point of SPap admits an etaleneighborhood U Ñ SPap in which U is a scheme and the morphism

ZPappmqU Ñ U

restricts to a closed immersion of schemes on every connected component

Z Ă ZPappmqU .

We will construct a Cartier divisor on any such U , and then glue themtogether as U varies over an etale cover to obtain the divisor YPappmq.


Fix Z as above, let I Ă OU be its ideal sheaf, and let Z 1 be the closedsubscheme of U defined by the ideal sheaf I2. Thus we have closed immer-sions

Z Ă Z 1 Ă U,

the first of which is a square-zero thickening.By the very definition of ZPappmq, along Z there is a universal Ok-linear

map x : A0Z Ñ AZ . This map does not extend to a map A0Z1 Ñ AZ1 ,however, by deformation theory [Lan13, Chapter 2.1.6] the induced Ok-linear morphism of vector bundles

x : HdR1 pA0Zq Ñ HdR

1 pAZq

admits a canonical extension to

(2.5.2) x1 : HdR1 pA0Z1q Ñ HdR

1 pAZ1q.

Recalling the morphism (2.4.3), define Y Ă Z 1 as the largest closed sub-scheme over which the composition

(2.5.3) HdR1 pA0Z1q bH

dR1 pA0Z1q

x1bx1ÝÝÝÑ HdR

1 pAZ1q bHdR1 pAZ1q

PÝÑ P|Z1

vanishes.

Lemma 2.5.4. If U Ñ SPap factors through SPap r Sing, then Y “ Z 1.

Proof. Proposition 2.4.3 provides us with a commutative diagram

HdR1 pA0Z1q

b2 x1bx1 //

p2.5.3q

,,

HdR1 pAZ1q

b2 qbq //`

LiepAZ1qFAZ1˘b2

–

P|Z1 ,

where

FAZ1 “ kerpε : LiepAZ1q Ñ LiepAZ1qq

as in Theorem 2.3.2.By deformation theory, Z Ă Z 1 is characterized as the largest closed

subscheme over which (2.5.2) respects the Hodge filtrations. Using (2.4.2),it is easily seen that Z Ă Z 1 can also be characterized as the largest closedsubscheme over which

H1pA0Z1qq˝x1ÝÝÑ LiepAZ1qFAZ1

vanishes identically. As Z Ă Z 1 is a square zero thickening, it follows firstthat the horizontal composition in the above diagram vanishes identically,and then that (2.5.3) vanishes identically. In other words Y “ Z 1.

Lemma 2.5.5. The closed subscheme Y Ă U is defined locally by one equa-tion.


Proof. Fix a closed point y P Y of characteristic p, let OU,y be the local ringof U at y, and let m Ă OU,y be the maximal ideal. For a fixed k ą 0, let

U “ SpecpOU,ymkq Ă U

be the kth-order infinitesimal neighborhood of y in U . The point of passingto the infinitesimal neighborhood is that p is nilpotent in OU , and so wemay apply Grothendieck-Messing deformation theory.

By construction we have closed immersions

Y

Z // Z 1 // U.

Applying the fiber product ˆUU throughout the diagram, we obtain closedimmersions

Y

Z // Z 1 // U

of Artinian schemes. As k is arbitrary, it suffices to prove that Y Ă U isdefined by one equation.

First suppose that p - D. In this case U Ñ U Ñ SPap factors throughthe nonsingular locus (2.3.2). It follows from Remark 2.5.1 that Z Ă U isdefined by one equation, and Z 1 is defined by the square of that equation.By Lemma 2.5.4, Y Ă U is also defined by one equation.

For the remainder of the proof we assume that p | D. In particular p ą 2.Consider the closed subscheme Z2 ãÑ U with ideal sheaf I3, so that we haveclosed immersions Z Ă Z 1 Ă Z2 Ă U. Taking the fiber product with U , theabove diagram extends to

Y

Z // Z 1 // Z2 // U .

As p ą 2, the cube zero thickening Z Ă Z2 admits divided powers ex-tending the trivial divided powers on Z Ă Z 1. Therefore, by Grothendieck-Messing theory, the restriction of (2.5.2) to

x1 : HdR1 pA0Z1q Ñ HdR

1 pAZ1q.

admits a canonical extension to

x2 : HdR1 pA0Z2q Ñ HdR

1 pAZ2q.

Define Y 1 Ă Z2 as the largest closed subscheme over which

(2.5.4) HdR1 pA0Z2q bH

dR1 pA0Z2q

x2bx2ÝÝÝÝÑ HdR

1 pAZ2q bHdR1 pAZ2q

PÝÑ P|Z2


vanishes identically, so that there are closed immersions

Y

// Y 1

Z // Z 1 // Z2 // U .

As in the proof of Lemma 2.5.4, we may characterizeZ Ă Z2 as the largestclosed subscheme along which x2 respects the Hodge filtrations. Equiva-lently, by (2.4.2), Z Ă Z2 is the largest closed subscheme over which thecomposition

HdR1 pA0Z2q

x2˝εÝÝÝÑ HdR

1 pAZ2qqÝÑ LiepAZ2q

vanishes identically. This implies that Z 1 Ă Z2 is the largest closed sub-scheme over which

(2.5.5) HdR1 pA0Z2q

b2 px2˝εqb2

ÝÝÝÝÝÑ HdR1 pAZ2q

qb2

ÝÝÑ LiepAZ2qb2

vanishes identically.We claim that Y “ Y 1. It follows directly from the definitions that

Y “ Y 1 X Z 1, and hence it suffices to show that Y 1 Ă Z 1. In other words,it suffices to shows that the vanishing of (2.5.4) implies the vanishing of(2.5.5). For local sections a and b of H1pAZ2q, define

Qabb : FilpAZ2q bľn´1

LiepAZ2q Ñľn

LiepAZ2q

by

Qabbpe1 b qpe2q ^ ¨ ¨ ¨ ^ qpenqq “ ψpa, e1q ¨ qpbq ^ qpe2q ^ ¨ ¨ ¨ ^ qpenq.

It is clear that Qabb depends on the images of a and b in LiepAZ2q, and thatthis construction defines an isomorphism(2.5.6)

LiepAZ2qb2 QÝÑ Hom

´

Fil0HdR1 pAZ2q b

ľn´1LiepAZ2q,

ľnLiepAZ2q

¯

.

It is related to the map

LiepAZ2qb2 PÝÑ Hom

´

ľnHdR

1 pAZ2qεHdR1 pAZ2q,

ľnLiepAZ2q

¯

of Proposition 2.4.3 by

Pabbpe1 ^ ¨ ¨ ¨ ^ enq “ Qεabεbpe1 b qpe2q ^ ¨ ¨ ¨ ^ qpenqq

for any local section e1 b e2 b ¨ ¨ ¨ b en of

Fil0HdR1 pAZ2q bH

dR1 pAZ2q b ¨ ¨ ¨ bH

dR1 pAZ2q.

Putting everything together, if (2.5.4) vanishes, then Px2pa0qbx2pb0q “ 0 for

all local sections a0 and b0 of HdR1 pA0Z2q. Therefore

Qx2pεa0qbx2pεb0q “ 0

for all local sections a0 and b0, which implies, as (2.5.6) is an isomorphism,that (2.5.5) vanishes. This proves that Y 1 Ă Z 1, and hence Y “ Y 1.


The map (2.5.4), whose vanishing defines Y 1 Ă Z2, factors through amorphism of line bundles

HdR1 pA0Z2qεH

dR1 pA0Z2q bH

dR1 pA0Z2qεH

dR1 pA0Z2q Ñ P|Z2 ,

and hence Y Ă Z2 is defined by one equation. In other words, if we denoteby I Ă OU and J Ă OU the ideal sheaves of Z Ă U and Y Ă U ,respectively, then I3 is the ideal sheaf of Z2 Ă U , and

J “ pfq ` I3

for some f P OU . But Y Ă Z 1 implies that I2 Ă J , and hence I3 Ă IJ .It follows that the image of f under the composition

J I3 Ñ J IJ Ñ J mJis an OU -module generator, and J is principal by Nakayama’s lemma.

For each connected component Z Ă ZPappmqU we have now defined aclosed subscheme Y Ă Z 1. By Lemma 2.5.5 it is an effective Cartier divisor,and summing these Cartier divisors as Z varies over all connected compo-nents yields an effective Cartier divisor YPappmqU on U . Letting U vary overan etale cover and applying etale descent defines a effective Cartier divisorYPappmq on SPap.

Proof of Theorem 2.5.2. The Cartier divisor YPappmq just defined agreeswith the square of ZPappmq on SPap r Sing. This is clear from Lemma2.5.4 and the definition of YPappmq. The uniqueness claim follows from thenormality of SPap, exactly as in the proof of Theorem 2.4.2.

2.6. Pullbacks of Cartier divisors. After Theorem 2.4.2 we have twoline bundles ΩKra and ω on SKra, which agree over the complement of theexceptional locus Exc. We wish to pin down more precisely the relationbetween them.

Similarly, after Theorem 2.5.2 we have Cartier divisors YKrapmq and2ZKrapmq. These agree on the complement of Exc, and again we wish topin down more precisely the relation between them.

Denote by π0pSingq the set of connected components of the singular locusSing Ă SPap. For each s P π0pSingq there is a corresponding irreducibleeffective Cartier divisor

Excs “ ExcˆSPaps ãÑ SKra

supported in a single characteristic dividing D. These satisfy

Exc “ğ

sPπ0pSingq

Excs.

Remark 2.6.1. As Sing is a reduced 0-dimensional stack of finite type overOkδOk, each s P π0pSingq can be realized as the stack quotient

s – GszSpecpFsqfor a finite field Fs of characteristic p | D acted on by a finite group Gs.


Fix a geometric point SpecpFq Ñ s. By mild abuse of notation thisgeometric point will again be denoted simply by s. It determines a pair

(2.6.1) pA0,s, Asq P SPappFq,and hence a positive definite hermitian Ok-module

Ls “ HomOkpA0,s, Asq

as in (2.5.1). This hermitian lattice depends only on s P π0pSingq, not onthe choice of geometric point above it.

Proposition 2.6.2. For each s P π0pSingq the abelian varieties A0s andAs are supersingular, and there is an Ok-linear isomorphism of p-divisiblegroups

(2.6.2) Asrp8s – A0srp

8s ˆ ¨ ¨ ¨ ˆA0srp8s

looooooooooooooomooooooooooooooon

n times

identifying the polarization on the left with the product polarization on theright. Moreover, the hermitian Ok-module Ls is self-dual of rank n.

Proof. Recall that p “ charpFq is odd and ramified in k. Denote by p Ă Ok

be the unique prime above it. Let W “ W pFq be the Witt ring of F, letFr P AutpW q be the unique continuous lift of the p-power Frobenius on F.

Certainly A0s is supersingular, as p is ramified in Ok Ă EndpA0sq.Let DpW q denote the covariant Dieudonne module of As, endowed with

its operators F and V satisfying FV “ p “ V F . The Dieudonne module isfree of rank n over Ok bZ W , and the short exact sequence

0 Ñ Fil0HdR1 pAsq Ñ HdR

1 pAsq Ñ LiepAsq Ñ 0

of F-modules is identified with

0 Ñ V DpW qpDpW q Ñ DpW qpDpW q Ñ DpW qV DpW q Ñ 0.

The element δ P Ok fixed in §1.7 satisfies ordppδq “ 1, and

δ ¨ DpW q “ V DpW q.Indeed, by Theorem 2.3.1 the Lie algebra LiepAsq is annihilated by δ, andhence δ ¨ DpW q Ă V DpW q. Equality holds as

dimF`

DpW qδ ¨ DpW q˘

“ n “ dimF`

DpW qV DpW q˘

.

Denote by N Ă DpW q the set of fixed points of the Fr-semilinear bijection

δ ˝ V ´1 : DpW q Ñ DpW q.It is a free Ok,p-module of rank n endowed with an isomorphism

DpW q – N bZp W

identifying V “ δbFr´1. Moreover, the alternating form ψ on DpW q inducedby the polarization on As has the form

ψpn1 b w1, n2 b w2q “ w1w2 ¨ TrkQ

ˆ

hpn1, n2q

δ

˙


for a perfect hermitian pairing h : N ˆ N Ñ Ok,p. By diagonalizing thishermitian form, we obtain an orthogonal decomposition of N into rank onehermitian Ok,p-modules, and tensoring this decomposition with W yields adecomoposition of DpW q as a direct sum of principally polarized Dieudonnemodules, each of height 2 and slope 12. This corresponds to a decomposi-tion (2.6.2) on the level of p-divisible groups.

In particular, As is supersingular, and hence is isogenous to n copies ofA0s. Using the Noether-Skolem theorem, this isogeny may be chosen to beOk-linear. It follows first that Ls has Ok-rank n, and then that the naturalmap

Ls bZ Zq – HomOkpA0srq

8s, Asrq8sq

is an isomorphism of hermitian Ok,q-modules for every rational prime q. Itis easy to see, using (2.6.2) when q “ p, that the hermitian module on theright is self-dual, and hence the same is true of each Ls bZ Zq.

The remainder of §2.6 is devoted to proving the following result.

Theorem 2.6.3. There is an isomorphism

ω2 bOpExcq – ΩKra

of line bundles on SKra, as well as an equality

2ZKrapmq “ YKrapmq `ÿ

sPπ0pSingq


of Cartier divisors.

Proof. Recall from Proposition 2.4.3 the morphism

ω2 ΩKra

LiepA0qb2 b pLiepAqFAqqb2

p2.4.4q // LiepA0qb2 b PKra,

whose restriction to SKrarExc is an isomorphism. If we view this morphismas a global section

(2.6.3) σ P H0pSKra,ω´2 bΩKraq,

then

(2.6.4) divpσq “ÿ

sPπ0pSingq

`sp0q ¨ Excs

for some integers `sp0q ě 0, and hence

(2.6.5) ω´2 bΩKra –â

sPπ0pSingq

OpExcsqb`sp0q.

We must show that each `sp0q “ 1.


Similarly, suppose m ą 0. It follows from Theorem 2.5.2 that

(2.6.6) 2ZKrapmq “ YKrapmq `ÿ

sPπ0pSingq

`spmq ¨ Excs

for some integers `spmq. Moreover, it is clear from the construction ofYKrapmq that 2ZKrapmq ´ YKrapmq is effective, and so `spmq ě 0. We mustshow that

`spmq “ #tx P Ls : xx, xy “ mu.

Fix s P π0pSingq, and let SpecpFq Ñ s, and pA0s, Asq P SPappFq be asin (2.6.1). The rational prime p “ charpFq is odd and ramified in k. LetW “ W pFq be the Witt ring of F, and set W “ Ok bZ W . It is a completediscrete valuation ring of absolute ramification degree 2. Fix a uniformizer$ PW. As p is odd, the quotient map

W ÑW$W “ Fadmits canonical divided powers.

Denote by D0 and D the Grothendieck-Messing crystals of A0s and As,respectively. Evaluation of the crystals3 along the divided power thickeningW Ñ F yields free Ok bZ W-modules D0pWq and DpWq endowed withalternating W-bilinear forms ψ0 and ψ, and Ok-linear isomorphisms

D0pWq$D0pWq – D0pFq – HdR1 pA0sq.

andDpWq$DpWq – DpFq – HdR

1 pAsq.

The W -modules D0pW q and DpW q are canonically identified with thecovariant Dieudonns modules of A0s and As, respectively. The operatorsF and V on these Dieudonne modules induce operators, denoted the sameway, on

D0pWq – D0pW q bW W, DpWq – DpW q bW W.

For any elements y1, . . . , yk in an OkbZW-module, let xy1, . . . , yky be theOk bZ W-submodule generated by them. Recall from §1.7 the elements

ε, ε P Ok bZ W.

Lemma 2.6.4. There is an Ok bZ W-basis e0 P D0pWq such that

FilD0pWqdef“ xεe0y Ă D0pWq

is a totally isotropic W-module direct summand lifting the Hodge filtrationon D0pFq, and such that V e0 “ δe0.

Similarly, there is an Ok bZ W-basis e1, . . . , en P DpWq such that

FilDpWq def“ xεe1, εe2, . . . , εeny Ă DpWq

3If p “ 3, the divided powers on W Ñ F are not nilpotent, and so we cannot evaluatethe usual Grothendieck-Messing crystals on this thickening. However, Proposition 2.6.2implies that the p-divisible groups of A0s and As are formal, and Zink’s theory of displays[Zin02] can be used as a substitute.


is a totally isotropic W-module direct summand lifting the Hodge filtrationon DpFq. This basis may be chosen so that so that V ek`1 “ δek, where theindices are understood in ZnZ, and also so that

ψ`

xeiy, xeiy˘

“

#

W if i “ j

0 otherwise.

Proof. As in the proof of Proposition 2.6.2, we may identify

D0pW q – N0 bZp W

for some free Ok,p-module N0 of rank 1, in such a way that V “ δ b Fr´1,and the alternating form on D0pW q arises as the W -bilinear extension ofan alternating form ψ0 on N0. Any Ok,p-generator e0 P N0 determines agenerator of the Ok,p bZp W-module

D0pWq – N0 bZp W,

which, using (2.4.2) has the desired properties.Now set N “ N0‘¨ ¨ ¨‘N0 (n copies), so that, by Proposition 2.6.2, there

is an isomorphismDpW q – N bZp W

identifying V “ δ b Fr´1, and the alternating bilinear form on DpW q arisesfrom an alternating form ψ on N . Let Zpn Ă W be the ring of integers inthe unique unramified degree n extension of Qp, and fix an action

ι : Zpn Ñ EndOk,ppNq

in such a way that ψpιpαqx, yq “ ψpx, ιpαqyq for all α P Zpn .There is an induced decomposition

DpW q –à

kPZnZDpW qk,

whereDpW qk “ te P DpW q : @α P Zpn , ιpαq ¨ e “ Frkpαq ¨ eu

is free of rank one over Ok bZ W . Now pick any Zpn-module generatore P N , view it as an element of DpW q, and let ek P DpW qk be its projectionto the kth summand. This gives an OkbZW -basis e1, . . . , en P DpW q, whichdetermines an Ok bZ W-basis of DpWq with the required properties.

By the Serre-Tate theorem and Grothendieck-Messing theory, the lifts ofthe Hodge filtrations specified in Lemma 2.6.4 determine a lift

(2.6.7) pA0s, Asq P SPappWqof the pair pA0s, Asq. These come with canonical identifications

HdR1 pA0sq – D0pWq, HdR

1 pAsq – DpWqunder which the Hodge filtrations correspond to the filtrations chosen inLemma 2.6.4. In particular, the Lie algebra of As is

LiepAsq – DpWqFilDpWq “ xe1, e2, . . . , enyxεe1, εe2, . . . , εeny.


The W-module direct summand

FAs “ xe2, . . . , enyxεe2, . . . , εeny

satisfies Kramer’s condition (§2.3), and so determines a lift of (2.6.7) to

pA0s, Asq P SKrapWq.To summarize: starting from a geometric point SpecpFq Ñ s, we have

used Lemma 2.6.4 to construct a commutative diagram

(2.6.8) SpecpFq

// Excs

// s

SpecpWq // SKra

// SPap.

Lemma 2.6.5. The pullback of the map (2.4.4) via SpecpWq Ñ SKra van-ishes identically along the closed subscheme SpecpW$Wq, but not alongSpecpW$2Wq.

Proof. The W-submodule of

(2.6.9) LiepAsq – DpWqxεe1, εe2, . . . , εeny

generated by e1 is Ok-stable. The action of Ok bZ W on this W-line is via

Ok bZ W αbx ÞÑiW pαqxÝÝÝÝÝÝÝÝÝÑW

(where iW : Ok ÑW is the inclusion), and this map sends ε to a uniformizer

of W; see §1.7. Thus the quotient map q : DpWq Ñ LiepAsq satisfies qpεe1q “

$qpe1q up to multiplication by an element of Wˆ. It follows that

Pe1be1pe1 ^ ¨ ¨ ¨ ^ enq “ $ ¨ ψpεe1, e1q ¨ qpe1q ^ qpe2q ^ ¨ ¨ ¨ ^ qpenq

up to scaling by Wˆ.We claim that ψpεe1, e1q P Wˆ. Indeed, as qpe1q generates a W-module

direct summand of (2.6.9), there is some

x P FilDpWq “ xεe1, εe2, . . . , εeny Ă DpWqsuch that ψpx, e1q PWˆ. We chose our basis in Lemma 2.6.4 in such a waythat ψpεei, e1q “ 0 for i ą 1. It follows that ψpεe1, e1q is a unit, and hencethe same is true of ψpεe1, e1q “ ψpe1, εe1q “ ´ψpεe1, e1q.

We have now proved that

Pe1be1pe1 ^ ¨ ¨ ¨ ^ enq “ $ ¨ qpe1q ^ qpe2q ^ ¨ ¨ ¨ ^ qpenq

up to scaling by Wˆ, from which it follows that

Pe1be1pe1 ^ ¨ ¨ ¨ ^ enq Pľn

LiepAsq

is divisible by $, but not by $2.The quotient

HdR1 pAsqεH

dR1 pAsq – DpWqxεe1, . . . , εeny


is generated as a W-module by e1, . . . , en. From the calculation of theprevious paragraph, it now follows that Pe1be1 P PKra|SpecpWq is divisible

by $ but not by $2. The quotient

LiepAsqFAs – DpWqxεe1, e2, . . . , eny

is generated as a W-module by the image of e1, and we at last deduce that

P P Hom`

pLiepAqFAqb2,PKra

˘

|SpecpWq

is divisible by $ but not by $2.

Recall the global section σ of (2.6.3). It follows immediately from Lemma2.6.5 that its pullback via SpecpWq Ñ SKra has divisor SpecpW$Wq, andhence

SpecpWq ˆSKradivpσq “ SpecpW$Wq,

Comparison with (2.6.4) proves both that `sp0q “ 1, and that

(2.6.10) SpecpWq ˆSKraExcs “ SpecpW$Wq.

Recalling (2.6.5), this completes the proof that ω´2 bΩKra – OpExcq.Given any

x P Ls “ HomOkpA0s, Asq,

denote by kpxq the largest integer such that x lifts to a morphism

A0s bW Wp$kpxqq Ñ As bW Wp$kpxqq.

Lemma 2.6.6. As Cartier divisors on SpecpWq, we have

ZKrapmq ˆSKraSpecpWq “

ÿ

xPLsxx,xy“m

SpecpW$kpxqWq.

Proof. Each x P Ls with xx, xy “ m determines a geometric point

(2.6.11) pA0z, Az, xq P ZKrapmqpFq.

and surjective morphisms

OSKra,x

xx ##OZKrapmq,x W,

where OZKrapmq,x is the etale local ring at (2.6.11), OSKra,x is the etale localring at the point below it, and the arrow on the right is induced by the mapSpecpWq Ñ SKra of (2.6.8). There is an induced isomorphism of W-schemes

OZKrapmq,x bOSKra,xW –Wp$kpxqq,

and the claim follows by summing over x.


Lemma 2.6.7. As Cartier divisors on SpecpWq, we have

YKrapmq ˆSKraSpecpWq “

ÿ

xPLsxx,xy“m

SpecpW$2kpxq´1Wq.

Proof. Each x P Ls “ HomOkpA0s, Asq with xx, xy “ m induces a morphism

of crystals D0 Ñ D, and hence a map

D0pWqxÝÑ DpWq

respecting the F and V operators. By Grothendieck-Messing deformationtheory, the integer kpxq is characterized as the largest integer such that thecomposition

Fil0HdR1 pA0sq

Ă // HdR1 pA0sq

x // HdR1 pAsq

q // LiepAsq

εD0pWqĂ // D0pWq

x // DpWq // DpWqxεe1,εe2,...,εeny

.

vanishes modulo $kpxq. In other words the composition

HdR1 pA0sq

x˝εÝÝÑ HdR

1 pAsqqÝÑ LiepAsq

vanishes modulo $kpxq, but not modulo $kpxq`1.Using the bases of Lemma 2.6.4, we expand

xpe0q “ a1e1 ` ¨ ¨ ¨ ` anen

with a1, . . . , an P Ok bZ W. The condition that x respects V implies thata1 “ ¨ ¨ ¨ “ an. Let us call this common value a, so that

qpxpεe0qq “ ε ¨ qpae1 ` ¨ ¨ ¨ ` aenq “ aε ¨ qpe1q

in LiepAsq. By the previous paragraph, this element is divisible by $kpxq

but not by $kpxq`1, and so

(2.6.12) qpaεe1q “ $kpxqqpe1q

up to scaling by Wˆ.On the other hand, the submodule of LiepAsq generated by qpe1q is iso-

morphic to pOkbZWqxεy –W, and ε acts on this quotient by a uniformizerin W. Thus

(2.6.13) εqpe1q “ $qpe1q

up to scaling by Wˆ.Combining (2.6.12) and (2.6.13) shows that, up to scaling by Wˆ,

aε “ $kpxq´1ε


in the quotient pOk bZ Wqxεy. By the injectivity of the quotient mapxεy Ñ pOk bZ Wqxεy, this same equality holds in xεy Ă Ok bZ W. Usingthis and (2.6.12), we compute

Pxpe0qbxpe0qpe1 ^ ¨ ¨ ¨ ^ enq

“ ψpaεe1, e1q ¨ qpaεe1q ^ qpe2q ^ ¨ ¨ ¨ ^ qpenq

“ $2kpxq´1 ¨ ψpεe1, e1q ¨ qpe1q ^ qpe2q ^ ¨ ¨ ¨ ^ qpenq

“ $2kpxq´1 ¨ qpe1q ^ qpe2q ^ ¨ ¨ ¨ ^ qpenq

up to scaling by Wˆ. Here, as in the proof of Lemma 2.6.5, we have usedψpεe1, e1q PWˆ.

This calculation shows that the composition

HdR1 pA0sq

b2 xbx // HdR1 pAsq

b2 P // P|SpecpWq

vanishes modulo $2kpxq´1, but not modulo $2kpxq, and the remainder of theproof is the same as that of Lemma 2.6.6: Comparing with the definition ofYKrapmq, see especially (2.5.3), shows that

OYKrapmq,x bOSKra,xW –Wp$2kpxq´1q,

and summing over all x proves the claim.

Combining Lemmas 2.6.6 and 2.6.7 shows that

SpecpWq ˆSKra

`

2ZKrapmq ´ YKrapmq˘

“ÿ

xPLsxx,xy“m

SpecpW$Wq

as Cartier divisors on SpecpWq. We know from (2.6.10) that

SpecpWq ˆSKraExct “

#

SpecpW$Wq if t “ s

0 if t ‰ s,

and comparison with (2.6.6) shows that

`spmq “ #tx P Ls : xx, xy “ mu,

completing the proof of Theorem 2.6.3.

3. Toroidal compactification

In this section we describe canonical toroidal compactifications

SKra//

S˚Kra

SPap

// S˚Pap,

and the structure of their formal completions along the boundary. Usingthis description, we define Fourier-Jacobi expansions of modular forms.


The existence of toroidal compactifications with reasonable properties isnot a new result. In fact the proof of Theorem 3.7.1, which asserts theexistence of a toroidal compactification with good properties, simply refersto [How15]. Of course [loc. cit.] is itself a very modest addition to theestablished literature [FC90, Lan13, Lar92, Rap78]. Because of this, thereader is perhaps owed a few words of explanation as to why §3 is so long.

It is well-known that the boundary charts used to construct toroidal com-pactifications of PEL-type Shimura varieties are themselves moduli spacesof algebro-geometric objects (either 1-motives, or, what is nearly the samething, degeneration data in the sense of [FC90]). This moduli interpretationis explained in §3.3.

It is a special feature of our particular Shimura variety ShpG,Dq that theboundary charts have a second, very different, moduli interpretation. Thissecond moduli interpretation is explained in §3.4. In some sense, the mainresult of §3 is not Theorem 3.7.1 at all, but rather Proposition 3.4.3, whichproves the equivalence of the two moduli problems.

If one bases the theory of Fourier-Jacobi coefficients on the second moduliinterpretation, it is easier to make explicit the relationship between Fourier-Jacobi coefficients of orthogonal modular forms and the Fourier-Jacobi co-efficients of their pullbacks to unitary modular forms. This will be essentialin §6 when we define a Borcherds product on ShpG,Dq as a pullback froman orthogonal Shimura variety, and then provide an algebraic interpretationof the leading Fourier-Jacobi coefficient of the pullback.

3.1. Cusp label representatives. The group G acts on both W0 and W .If J Ă W is an isotropic k-line, its stabilizer P “ StabGpJq is a parabolicsubgroup of G. This establishes a bijection between isotropic k-lines in Wand proper parabolic subgroups of G.

Definition 3.1.1. A cusp label representative for pG,Dq is a pair Φ “ pP, gqin which g P GpAf q and P Ă G is a parabolic subgroup. If P “ StabGpJqfor an isotropic k-line J Ă W , we call Φ a proper cusp label representative.If P “ G we call Φ an improper cusp label representative.

For each cusp label representative Φ “ pP, gq there is a distinguishednormal subgroup QΦ C P . If P “ G we simply take QΦ “ G. If P “

StabGpJq for an isotropic k-line J ĂW then, following the recipe of [Pin89,§4.7], we define QΦ as the fiber product

(3.1.1) QΦνΦ //

ReskQGm

aÞÑpa,Nmpaq,a,idq

P // GUpW0q ˆGLpJq ˆGUpJKJq ˆGLpW JKq.

The morphism GÑ GUpW q restricts to an injection QΦ ãÑ GUpW q, as theaction of QΦ on JKJ determines its action on W0.


Let K Ă GpAf q be the compact open subgroup (2.1.3). Any cusp labelrepresentative Φ “ pP, gq determines compact open subgroups

KΦ “ gKg´1 XQΦpAf q, KΦ “ gKg´1 X P pAf q,and a finite group

(3.1.2) ∆Φ “`

P pQq XQΦpAf qKΦ

˘

QΦpQq.

Definition 3.1.2. Two cusp label representatives Φ “ pP, gq and Φ1 “pP 1, g1q are K-equivalent if there exist γ P GpQq, h P QΦpAf q, and k P Ksuch that

pP 1, g1q “ pγPγ´1, γhgkq.

One may easily verify that this is an equivalence relation. Obviously, thereis a unique K-equivalence class of improper cusp label representatives.

From now through §3.6, we fix a proper cusp label representative Φ “

pP, gq, with P Ă G the stabilizer of an isotropic k-line J Ă W . There is aninduced weight filtration wtiW ĂW defined by

0 Ă J Ă JK Ă W

wt´3W Ă wt´2W Ă wt´1W Ă wt0W,

and an induced weight filtration on V “ HomkpW0,W q defined by

HomkpW0, 0q Ă HomkpW0, Jq Ă HomkpW0, JKq Ă HomkpW0,W q

wt´2V Ă wt´1V Ă wt0V Ă wt1V ,

It is easy to see that wt´1V is an isotropic k-line, whose orthogonal withrespect to (2.1.5) is wt0V . Denote by griW “ wtiW wti´1W the gradedpieces, and similarly for V .

The Ok-lattice ga ĂW determines an Ok-lattice

gripgaq “`

gaX wtiW˘

`

gaX wti´1W˘

Ă griW.

The outer graded pieces

(3.1.3) m “ gr´2pgaq, n “ gr0pgaq

are projective rank one Ok-modules. Each of these Ok-modules is deter-mined by the other, as the hermitian form on ga induces a perfect hermitianpairing4

mˆ nÑ Ok.

The middle graded piece gr´1pgaq is endowed with a positive definite self-dual hermitian form, inherited from the self-dual form on ga appearing inthe proof of Proposition 2.1.1.

4This implies that m – n, but identifying them can only lead to confusion.


Remark 3.1.3. The isometry class of ga as a hermitian lattice is determinedby the isomorphism classes of m and n, and the isometry class of gr´1pgaq.This follows from the proof of [How15, Proposition 2.6.3], which shows thatone can find an isomorphism5 of hermitian lattices

ga – gr´2pgaq ‘ gr´1pgaq ‘ gr0pgaq,

where the right hand side is given the unique self-dual hermitian form re-stricting to the given form on the middle summand, and for which the outersummands are totally isotropic and orthogonal to the middle summand.

Exactly as in (2.1.4), there is a k-conjugate linear isomorphism

HomkpW0, gr´1W qx ÞÑx_ÝÝÝÝÑ Homkpgr´1W,W0q.

If we define

L0 “ HomOkpga0, gr´1pgaqq(3.1.4)

Λ0 “ HomOkpgr´1pgaq, ga0q,

then x ÞÑ x_ restricts to an Ok-conjugate linear isomorphism L0 – Λ0. Weendow L0 with the positive definite hermitian form

xx1, x2y “ x_1 ˝ x2 P EndOkpgaq – Ok

analogous to (2.1.5), and endow Λ0 with the “dual” hermitian form

xx_2 , x_1 y “ xx1, x2y.

Lemma 3.1.4. Two proper cusp label representatives Φ and Φ1 are K-equivalent if and only if Λ0 – Λ10 as hermitian Ok-modules, and n – n1

as Ok-modules. Moreover, the finite group (3.1.2) satisfies

(3.1.5) ∆Φ – UpΛ0q ˆGLOkpnq.

Proof. The first claim is an elementary exercise, left to the reader. Forthe second claim we only define the isomorphism (3.1.5), and again leavethe details to the reader. The group P pQq acts on both W0 and W , pre-serving their weight filtrations, and so acts on both the hermitian spaceHomkpgr´1W,W0q and the k-vector space gr0W . The subgroup P pQq XQΦpAf qKΦ preserves the lattices

Λ0 Ă Homkpgr´1W,W0q

and n Ă gr0W , inducing (3.1.5).

5This uses our standing assumption that k has odd discriminant.


3.2. Mixed Shimura varieties. The subgroup QΦpRq Ă GpRq acts on

DΦpW q “ tk-stable R-planes y ĂW pRq : W pRq “ JKpRq ‘ yu,

and so also acts on

DΦ “ DpW0q ˆDΦpW q.

The hermitian domain of (2.1.2) satisifies DpW q Ă DΦpW q, and hence thereis a canonical QΦpRq-equivariant inclusion D Ă DΦ.

The mixed Shimura variety

(3.2.1) ShpQΦ,DΦqpCq “ QΦpQqzDΦ ˆQΦpAf qKΦ

admits a canonical model ShpQΦ,DΦq over k by the general results of [Pin89].By rewriting the double quotient as

ShpQΦ,DΦqpCq – QΦpQqzDΦ ˆQΦpAf qKΦKΦ,

we see that (3.2.1) admits an action of the finite group ∆Φ of (3.1.2), induced

by the action of P pQqXQΦpAf qKΦ on both factors of DΦˆQΦpAf qKΦ. Thisaction descends to an action on the canonical model.

Proposition 3.2.1. The morphism νΦ of (3.1.1) induces a surjection

ShpQΦ,DΦqpCqpz,hqÞÑνΦphqÝÝÝÝÝÝÝÝÑ kˆzpkˆ pOˆk

with connected fibers. This map is ∆Φ-equivariant, where ∆Φ acts triviallyon the target. In particular, the number of connected components of (3.2.1)is equal to the class number of k, and the same is true of its orbifold quotientby the action of ∆Φ.

Proof. The space DΦ is connected, and the kernel of νΦ : QΦ Ñ ReskQGm

is unipotent (so satisfies strong approximation). Therefore

π0

`

ShpQΦ,DΦqpCq˘

– QΦpQqzQΦpAf qKΦ – kˆzpkˆνΦpKΦq,

and an easy calculation shows that νΦpKΦq “ pOˆk .

It will be useful to have other interpretations of DΦ.

Remark 3.2.2. Any point y P DΦpW q determines a mixed Hodge structureon W whose weight filtration wtiW Ă W was define above, and whoseHodge filtration is defined exactly as in Remark 2.1.2. As in [PS08, p. 64]

or [Pin89, Proposition 1.2] there is an induced bigrading W pCq “À

W pp,qq,and this bigrading is induced by a morphism SC Ñ GUpW qC taking valuesin the stabilizer of JpCq. The product of this morphism with the morphismSC Ñ GUpW0qC of Remark 2.1.2 defines a map z : SC Ñ QΦC, and thisrealizes DΦ Ă HompSC, QΦCq.

Remark 3.2.3. Imitating the construction of Remark 2.1.3 identifies

DΦ –

w P εV pCq : V pCq “ wt0V pCq ‘ Cw ‘ Cw(

Cˆ.


3.3. The first moduli interpretation. Starting from the pair pΛ0, nq, wenow construct a smooth integral model of (3.2.1). Following the generalrecipes of the theory of arithmetic toroidal compactifications, as in [FC90,How15, Madb, Lan13], this integral model will be defined as the top layerof a tower of morphisms

CΦ Ñ BΦ Ñ AΦ Ñ SpecpOkq,

smooth of relative dimensions 1, n´ 2, and 0, respectively.Recall from §2.3 the smooth Ok-stack

Mp1,0q ˆOkMpn´2,0q Ñ SpecpOkq

of relative dimension 0 parametrizing certain pairs pA0, Bq of polarizedabelian schemes over S with Ok-actions. The etale sheaf HomOk

pB,A0q

on S is locally constant; this is a consequence of [BHY15, Theorem 5.1].Define AΦ as the moduli space of triples pA0, B, %q over Ok-schemes S, in

which pA0, Bq is an S-point of Mp1,0q ˆOkMpn´2,0q, and

% : Λ0 – HomOkpB,A0q

is an isomorphism of etale sheaves of hermitian Ok-modules.Define BΦ as the moduli space of quadruples pA0, B, %, cq over Ok-schemes

S, in which pA0, B, %q is an S-point of AΦ, and c : n Ñ B is an Ok-linearhomomorphism of group schemes over S. In other words, if pA0, B, %q is theuniversal object over AΦ, then

BΦ “ HomOkpn, Bq.

Suppose we fix µ, ν P n. For any scheme U and any morphism U Ñ

BΦ, there is a corresponding quadruple pA0, B, %, cq over U . Evaluating themorphism of U -group schemes c : n Ñ B at µ and ν determines U -pointscpµq, cpνq P BpUq, and hence determines a morphism of BΦ-schemes

UcpµqˆcpνqÝÝÝÝÝÝÑ B ˆB – B ˆB_.

Denote by Lpµ, νqU the pullback of the Poincare bundle via this morphism.As U varies, these line bundles are obtained as the pullback of a single

line bundle Lpµ, νq on BΦ, which depends, up to canonical isomorphism,only on the image of µb ν in

SymΦ “ Sym2Zpnq

@

pxµq b ν ´ µb pxνq : x P Ok, µ, ν P nD

.

Thus we may associate to every χ P SymΦ a line bundle Lpχq on BΦ, andthere are canonical isomorphisms

Lpχq b Lpχ1q – Lpχ` χ1q.Our assumption that D is odd implies that SymΦ is a free Z-module of rankone. Moreover, there is positive cone in SymΦbZR uniquely determined bythe condition µb µ ě 0 for all µ P n. Thus all of the line bundles Lpχq arepowers of the distinguished line bundle

(3.3.1) LΦ “ Lpχ0q


determined by the unique positive generator χ0 P SymΦ.At last, define BΦ-stacks

CΦ “ IsopLΦ,OBΦq, C˚Φ “ HompLΦ,OBΦ

q.

In other words, C˚Φ is the total space of the line bundle L´1Φ , and CΦ is the

complement of the zero section BΦ ãÑ C˚Φ. In slightly fancier language,

CΦ “ SpecBΦ

´

à

`PZL`Φ

¯

, C˚Φ “ SpecBΦ

´

à

`ě0

L`Φ¯

,

and the zero section BΦ ãÑ C˚Φ is defined by the ideal sheafÀ

`ą0 L`Φ.

Remark 3.3.1. Using the isomorphism of Lemma 3.1.4, the group ∆Φ actson BΦ via

pu, tq ‚ pA0, B, %, cq “ pA0, B, % ˝ u´1, c ˝ t´1q,

for pu, tq P UpΛ0q ˆ GLOkpnq. The line bundle LΦ is invariant under ∆Φ,

and hence the action of ∆Φ lifts to both CΦ and C˚Φ.

Proposition 3.3.2. There is a ∆Φ-equivariant isomorphism

ShpQΦ,DΦq – CΦk.

Proof. This is a special case of the general fact that mixed Shimura varietiesappearing at the boundary of PEL Shimura varieties are themselves modulispaces of 1-motives endowed with polarizations, endomorphisms, and levelstructure. The core of this is Deligne’s theorem [Del74, §10] that the cat-egory of 1-motives over C is equivalent to the category of integral mixedHodge structures of types p´1,´1q, p´1, 0q, p0,´1q, p0, 0q. See [Madb, §2],where this is explained in the case of Siegel modular varieties, and also[Bry83].

To make this a bit more explicit in our case, first recall that a 1-motiveA over a scheme S has an increasing weight filtration

0 “ wt´3A Ă wt´2A Ă wt´1A Ă wt0A “ A

by sub-1-motives, in which wt´1A is an extension

0 Ñ wt´2AÑ wt´1AÑ gr´1AÑ 0

of an abelian scheme by a torus. A good reference for 1-motives is [ABV05].Define XΦ to be the moduli stack over Ok whose S-valued points consist

of triples pA, ι, ψq in which

‚ A is a 1-motive over S,‚ ι : Ok Ñ EndpAq is an action,‚ ψ : AÑ A_ is a principal polarization, such that the induced Rosati

involution : on End0pAq satisfies ιpαq: “ ιpαq for all α P Ok,‚ the abelian scheme gr´1A, with its induced action of Ok, defines an

object of Mpn´2,0qpSq,‚ there is an Ok-linear isomorphism wt´2A – mbGm.


In the final condition, m is the rank one Ok-module of (3.1.3), and theisomorphism is part of the moduli problem. As usual we abbreviate such atriple to A P XΦpSq for simplicity.

Both k-stacks appearing in the statement of the proposition parametrizetriples pA0, A, %q over k-schemes S in which

‚ pA0, Aq PMp1,0qpSq ˆ XΦpSq,‚ % : Λ0 – HomOk

pgr´1A,A0q is an isomorphism of etale sheaves ofhermitian Ok-modules.

The fact that CΦk has this interpretation amounts to unpacking the defini-tion of a 1-motive, which we leave to the reader. To verify that ShpQΦ,DΦq

has this interpretation, one uses Remark 3.2.2 to interpret ShpQΦ,DΦqpCqas a moduli space of mixed Hodge structures on W0 and W , and uses thetheorem of Deligne cited above. The proof that the isomorphism descendsto the reflex field is identical to the proof for Siegel mixed Shimura varietiesfound in [Madb, §2.2].

We remark in passing that any triple pA0, A, %q as above automatically sat-isfies (2.2.4) for every prime `. Indeed, both sides of (2.2.4) are now endowedwith weight filtrations, analogous to the weight filtration on HomkpW0,W qdefined in §3.1. The isomorphism % induces an isomorphism (as hermitianOk,`-lattices) between the gr0 pieces on either side. The gr´1 and gr1 pieceshave no structure other then projective Ok,`-modules of rank 1, so are iso-morphic. These isomorphisms of graded pieces imply the existence of anisomorphism (2.2.4), exactly as in Remark 3.1.3.

3.4. The second moduli interpretation. In order to make explicit cal-culations, it will be useful to interpret the moduli spaces

CΦ Ñ BΦ Ñ AΦ Ñ SpecpOkq

in a different way.Suppose E Ñ S is an elliptic curve over any base scheme. If U is any

S-scheme and a, b P EpUq, we obtain an OU -module PEpa, bq by pullingback the Poincare bundle via

Upa,bqÝÝÝÑ E ˆS E – E ˆS E

_.

The notation is intended to remind the reader of the bilinearity propertiesof the Poincare bundle, as expressed by the canonical isomorphisms

PEpa` b, cq – PEpa, cq b PEpb, cq(3.4.1)

PEpa, b` cq – PEpa, bq b PEpa, cqPEpa, bq – PEpb, aq,

along with PEpe, bq – OU – PEpa, eq. Here e P EpUq is the zero section.Let E Ñ Mp1,0q be the universal elliptic curve with complex multipli-

cation by Ok. Its Poincare bundle satisfies, for all α P Ok, the additional


relation PEpαa, bq – PEpa, αbq. Abbreviate

E b L0 “ E bOkL0

for Serre’s tensor construction, and denote by PEbL0 the line bundle onpE b L0q ˆMp1,0q

pE b L0q whose pullback via a pair of U -valued points

c “ÿ

si b xi P EpUq b L0, c1 “ÿ

s1j b x1j P EpUq b L0

is the OU -module

PEbL0pc, c1q “

â

i,j

PEpxxi, x1jysi, s1jq.

Define QEbL0 to be the line bundle on E b L0 whose restriction to theU -valued point c “

ř

si b xi is

(3.4.2) QEbL0pcq “â

iăj

PEpxxi, xjysi, sjq bâ

i

PEpγxxi, xiysi, siq,

where

γ “1` δ

2P Ok.

It is related to PEbL0 by canonical isomorphisms

PEbL0pa, bq – QEbL0pa` bq bQEbL0paq´1 bQEbL0pbq

´1(3.4.3)

PEbL0pa, aq – QEbL0paqb2.

for all U -valued points a, b P EpUq b L0.

Remark 3.4.1. The line bundle PEbL0pδa, aq is canonically trivial. Thisfollows by comparing

PEbL0pγa, aqb2 – PEbL0pa, aq b PEbL0pδa, aq

with

PEbL0pγa, aqb2 – PEbL0pγa, aq b PEbL0pγa, aq – PEbL0pa, aq.

Remark 3.4.2. One should compare (3.4.3) with [Lan13, Construction 1.3.2.7]or [MFK94, Definition 6.2]. The line bundle QEbL0 determines a polariza-tion of E b L0, and PEbL0 is the pullback of the Poincare bundle via

pE b L0q ˆMp1,0qpE b L0q Ñ pE b L0q ˆMp1,0q

pE b L0q_.

Proposition 3.4.3. As above, let E ÑMp1,0q be the universal object. Thereare canonical isomorphisms

CΦ//

–

BΦ//

–

AΦ

–

IsopQEbL0 ,OEbL0q

// E b L0//Mp1,0q,

and the middle vertical arrow identifies LΦ – QEbL0.


Proof. Define a morphism AΦ ÑMp1,0q by sending a triple pA0, B, %q to theCM elliptic curve

(3.4.4) E “ HomOkpn, A0q.

To show that this map is an isomorphism we will construct the inverse.If S is any Ok-scheme and E P Mp1,0qpSq, we may define pA0, B, %q P

AΦpSq by setting

A0 “ E bOkn, B “ HomOk

pΛ0, A0q,

and taking for % : Λ0 – HomOkpB,A0q the tautological isomorphism. It

remains to endow B with a principal polarization. Let U be an S-scheme,and suppose we are given points

b, b1 P BpUq “ HomOkpΛ0, A0U q

of the form b “ x¨, λya and b1 “ x¨, λ1ya1 with λ, λ1 P Λ0 and a, a1 P A0pUq.Note that all points of BpUq are sums of points of this form. Define a linebundle

PBpb, b1q “ PA0pa, xλ, λ1ya1q

on U , and extend the definition Z-bilinearly, in the sense of (3.4.1), to arbi-trary points of BpUq. There is a unique line bundle PB on B ˆS B whoserestriction to any U -point pb, b1q P BpUq ˆ BpUq is PBpb, b1q, and a uniqueprincipal polarization on B such that PB is the pullback of the Poincarebundle via BˆS B Ñ BˆS B

_. The construction E ÞÑ pA0, B, %q is inverseto the above morphism AΦ ÑMp1,0q.

Now identify AΦ – Mp1,0q using the above isomorphism, and denoteby pA0, B, %q and E the universal objects. They are related by canonicalisomorphisms

HomOkpnbOk

Λ0, A0q

–

tt

–

))BΦ “ HomOk

pn, Bq HomOkpΛ0, Eq.

Combining this with the Ok-linear isomorphism

E b L0abx ÞÑx ¨ ,x_yaÝÝÝÝÝÝÝÝÝÑ HomOk

pΛ0, Eq

defines BΦ – E b L0. All that remains is to prove that this isomorphismidentifies LΦ with QEbL0 .

Any fractional ideal b Ă k admits a unique positive definite self-dualhermitian form, given explicitly by xb1, b2y “ b1b2Npbq. It follows that anyrank one projective Ok-module admits a unique positive definite self-dualhermitian form. For the Ok-module HomOk

pn,Okq, this hermitian form is

x`1, `2y “ `1pµq`2pνq ` `1pνq`2pµq,

where µb ν “ χ0 P SymΦ is the positive generator appearing in (3.3.1).


The relation (3.4.4) implies a relation between the line bundles PE andPA0 . If U is any AΦ-scheme and we are given points

s, s1 P EpUq “ HomOkpn, A0U q

of the form s “ `p¨qa and s1 “ `1p¨qa1 with `, `1 P HomOkpn,Okq and a, a1 P

A0pUq, then

PEps, s1q – PA0

`

x`, `1ya, a1˘

PEpγs, sq – PA0

`

`pµqa, `pνqa˘

.

Fix any point c P BΦpUq. Using the isomorphisms in the diagram above,c admits three different interpretations. In one of them, c has the form

c “ÿ

ìp¨qx¨, λiyai P HomOkpnbOk

Λ, A0U q.

By setting

bi “ x¨, λiyai P HomOkpΛ0, A0U q “ BpUq

si “ ìp¨qai P HomOkpn, A0U q “ EpUq,

we find the other two interpretations

c “ÿ

ìp¨qbi P HomOkpn, BU q

c “ÿ

x¨, λiysi P HomOkpΛ0, EU q.

The above relations between PB, PE , and PA0 imply

PBpcpµq, cpνqq–

â

i,j

PBpìpµqbi, `jpνqbjq

–â

i,j

PA0pìpµqai, xλi, λjy`jpνqajq

–â

iăj

PA0pxì, `jyai, xλi, λjyajq bâ

i

PA0pìpµqai, ìpνqxλi, λiyaiq

–â

iăj

PEpsi, xλi, λjysjq bâ

i

PEpγsi, xλi, λiysiq.

Now write λi “ x_i with xi P L0, and use the relation

PEpsi, xλi, λjysjq “ PEpxλj , λiysi, sjq “ PEpxxi, xjysi, sjq

to obtain an isomorphism PBpcpµq, cpνqq – QEbL0pcq. The line bundle onthe left is precisely the pullback of LΦ via c, and letting c vary we obtainan isomorphism LΦ – QEbL0 .


3.5. The line bundle of modular forms. We now define a line bundle ofweight one modular forms on our mixed Shimura variety, analogous to theone on the pure Shimura variety defined in §2.4.

The holomorphic line bundle ωan on D defined in §2.4 admits a canonicalextension to DΦ, denoted ωanΦ , whose fiber at a point z “ py0, yq is againdefined by (2.4.1). If we embed DΦ into projective space over εV pCq as inRemark 3.2.3, this is simply the restriction of the tautological bundle. Thereis an obvious action of QΦpRq on the total space of ωanΦ , lifting the naturalaction on DΦ, and so ωanΦ descends to a holomorphic line bundle on thecomplex orbifold ShpQΦ,DΦqpCq.

As in §2.4, the holomorphic line bundle ωanΦ is algebraic and descends tothe canonical model ShpQΦ,DΦq. In fact, it admits a canonical extension tothe integral model CΦ, as we now explain.

Again denote by E ÑMp1,0q the universal CM elliptic curve. Using theisomorphism AΦ –Mp1,0q of Proposition 3.4.3, define a line bundle

(3.5.1) ωΦ “ LiepEq´1

on AΦ, and denote in the same way its pullback to any step in the tower

C˚Φ Ñ BΦ Ñ AΦ.

Proposition 3.5.1. If pA0, B, %q denotes the universal object over AΦ, then

ωΦ – LiepA0q´1 bOk

n.

Moreover, the isomorphism CΦpCq – ShpQΦ,DΦqpCq of Proposition 3.3.2identifies the analytification of ωΦ with the already defined ωanΦ .

Proof. The first claim is clear from (3.4.4). For the second claim, recall thatthe isomorphism CΦpCq – ShpQΦ,DΦqpCq of Proposition 3.3.2 is obtainedby identifying both spaces with the moduli space of triples pA0, A, %q, where

pA0, Aq PMp1,0qpCq ˆ XΦpCq,

and the third datum % is irrelevant to the proof at hand.Let pA0, A, %q denote the universal such triple. The 1-motive A has a de

Rham realization HdR1 pAq. It is a locally free sheaf of rank 2n on CΦpCq,

endowed with an Ok-stable Hodge filtration F 0HdR1 pAq Ă HdR

1 pAq. The Liealgebra

LiepAq “ HdR1 pAqF 0HdR

1 pAq

is locally free of rank n. Recall that m b Gm – wt´2A. This implies thatthere is an exact sequence

0 Ñ LiepmbGmq Ñ LiepAq Ñ Liepgr´1Aq Ñ 0,

where every fiber of the abelian scheme gr´1A Ñ CΦpCq is a point ofMpn´2,0qpCq. The signature condition implies that ε kills on the Lie algebraof gr´1A, yielding an isomorphism

ε ¨ LiepmbGmq – ε ¨ LiepAq


of line bundles on CΦpCq.Let m denote the Z-module m of (3.1.3), but endowed with its conjugate

Ok-module structure. The hermitian pairing between m and n defines anisomorphism n bOk

m – Ok. Thus we may rewrite the isomorphism of thefirst claim as

ω´1Φ – LiepA0q bOk

m

– LiepA0q b εLiepmbGmq

– LiepA0q b εLiepAq

– LiepA0q b`

εFil0HdR1 pAq

˘´1,

where the final isomorphism arises from the perfect pairing (2.2.1) inducedby the principal polarization of A.

The above isomorphisms identify

ωΦ – Hom`

LiepA0q, εFil0HdR1 pAq

˘

– Hom`

HdR1 pA0qεH

dR1 pA0q, εFil0HdR

1 pAq˘

.

Tracing through Deligne’s explicit realization of the category of 1-motivesover C as a category of mixed Hodge structures, one can see that the fi-nal expression agrees with the holomorphic line bundle on ShpQΦ,DΦqpCqdefined by (2.4.1).

3.6. Special divisors. Let Y0pDq be the moduli stack over Ok parametriz-ing cyclic D-isogenies of elliptic curves over Ok-schemes, and let E Ñ E 1 bethe universal object.

Let pA0, B, %, cq be the universal object over BΦ. Recalling the Ok-conjugate linear isomorphism L0 – Λ0 defined after (3.1.4), each x P L0

defines a morphism

ncÝÑ B

%px_qÝÝÝÑ A0

of sheaves of Ok-modules on BΦ. Define ZΦpxq Ă BΦ as the largest closedsubstack over which this morphism is trivial. We will see in a moment thatthis closed substack is defined locally by one equation. For any m ą 0 definestack over BΦ by

(3.6.1) ZΦpmq “ğ

xPL0xx,xy“m

ZΦpxq.

We also view ZΦpmq as a divisor on BΦ, and denote in the same way thepullback of this divisor via C˚Φ Ñ BΦ.

We will now reformulate the definition of ZΦpxq in terms of the moduliproblem of §3.4. Recalling the isomorphisms of Proposition 3.4.3, every


x P L0 determines a commutative diagram

BΦ– //

E b L0x¨,xy //

E //

E

AΦ

– //Mp1,0q Mp1,0q// Y0pDq,

where Mp1,0q Ñ Y0pDq endows E with the cyclic D-isogeny δ : E Ñ E, andthe rightmost square is cartesian. The upper and lower horizontal composi-tions are denoted jx and j, giving the diagram

(3.6.2) BΦjx //

E

AΦ

j // Y0pDq.

Proposition 3.6.1. For any nonzero x P L0, the closed substack ZΦpxq ĂBΦ is equal to the pullback of the zero section along jx. It is an effectiveCartier divisor, flat over AΦ. In particular, as AΦ is flat over Ok, so iseach divisor ZΦpxq.

Proof. Recall the isomorphisms

E – HomOkpn, A0q, B – HomOk

pΛ0, A0q

from the proof of Proposition 3.4.3. If we identify A0 bOkL0 – B using

A0 bOkL0

abx ÞÑx¨,x_yaÝÝÝÝÝÝÝÝÑ HomOk

pΛ0, A0q – B,

we obtain a commutative diagram of AΦ-stacks

E bOkL0

//

x¨,xy

HomOkpn, A0 bOk

L0q // HomOkpn, Bq “ BΦ

%px_q

E // HomOk

pn, A0q,

in which all horizontal arrows are isomorphisms. The first claim followsimmediately.

The remaining claims now follow from the cartesian diagram

ZΦpxq //

Mp1,0q

e

BΦ

– // E b L0x¨,xy // E.

The zero section e : Mp1,0q ãÑ E is locally defined by one equation [KM85,Lemma 1.2.2], and so the same is true of its pullback ZΦpxq ãÑ BΦ. Com-position along the bottom row is flat by [MFK94, Lemma 6.12], and henceso is the top horizontal arrow.


Remark 3.6.2. For those who prefer the language of 1-motives: As in theproof of Proposition 3.3.2, there is a universal triple pA0, A, %q over CΦ inwhich A0 is an elliptic curve with Ok-action and A is a principally polarized1-motive with Ok-action. The functor of points of ZΦpmq assigns to anyscheme S Ñ CΦ the set

ZΦpmqpSq “ tx P HomOkpA0,S , ASq : xx, xy “ mu,

where the positive definite hermitian form x¨, ¨y is defined as in (2.5.1). Whenexpressed this way, our special divisors are obviously analogous to the specialdivisors on SKra defined in §2.5.

3.7. The toroidal compactification. We describe the canonical toroidalcompactification of the integral model SKra from §2.3.

Theorem 3.7.1. Let S denote either SKra or SPap. There is a canonicaltoroidal compactification S ãÑ S˚, flat over Ok of relative dimension n´1.It admits a stratification

S˚ “ğ

Φ

S˚pΦq

as a disjoint union of locally closed substacks, indexed by the K-equivalenceclasses of cusp label representatives.

(1) The stack S˚Kra is regular.(2) The stack S˚Pap is Cohen-Macaulay and normal, with Cohen-Macaulay

and geometrically normal fibers.(3) The open dense substack S Ă S˚ is the stratum indexed by the

unique equivalence class of improper cusp label representatives. Itscomplement

BS˚ “ğ

Φ proper

S˚pΦq

is a smooth divisor, flat over Ok.(4) For each proper Φ the stratum S˚pΦq is closed. All components of

S˚pΦqC are defined over the Hilbert class field H, and they are per-muted simply transitively by GalpHkq. Moreover, there is a canon-ical identification of Ok-stacks

∆ΦzBΦ

S˚pΦq

∆ΦzC˚Φ S˚

such that the two stacks in the bottom row become isomorphic aftercompletion along their common closed substack in the top row. Inother words, there is a canonical isomorphism of formal stacks

(3.7.1) ∆ΦzpC˚Φq^BΦ– pS˚q^S˚pΦq.


(5) The morphism SKra Ñ SPap extends uniquely to a stratum preservingmorphism of toroidal compactifications. This extension restricts toan isomorphism

(3.7.2) S˚Kra r Exc – S˚Pap r Sing,

compatible with (3.7.1) for any proper Φ.(6) The line bundle ω on SKra defined in §2.4 admits a unique extension

(denoted the same way) to the toroidal compactification in such a waythat (3.7.1) identifies it with the line bundle ωΦ on C˚Φ. A similarstatement holds for ΩKra, and these two extensions are related by

ΩKra – ω2 bOpExcq.

(7) The line bundle ΩPap on SPap defined in §2.4 admits a unique ex-tension (denoted the same way) to the toroidal compactification, insuch a way that (3.7.1) identifies it with ω2

Φ.(8) For any m ą 0, define Z˚Krapmq as the Zariski closure of ZKrapmq in

S˚Kra. The isomorphism (3.7.1) identifies it with the Cartier divisorZΦpmq on C˚Φ.

(9) For any m ą 0, define Y˚Pappmq as the Zariski closure of YPappmq

in S˚Pap. The isomorphism (3.7.1) identifies it with the square of the

Cartier divisor ZΦpmq. Moreover, the pullback of Y˚Pappmq to S˚Kra,

denoted Y˚Krapmq, satisfies

2 ¨ Z˚Krapmq “ Y˚Krapmq `ÿ

sPπ0pSingq

#tx P Ls : xx, xy “ mu ¨ Excs.

Proof. Briefly, in [How15, §2] one finds the construction a canonical toroidalcompactification

Mpn´1,1q ãÑM,˚

pn´1,1q.

Using the open and closed immersion

S ãÑMp1,0q ˆMpn´1,1q,

the toroidal compactification S˚ is defined as the Zariski closure of S in

Mp1,0q ˆM,˚pn´1,1q. All of the claims follow by examination of the construc-

tion of the compactification, along with Theorem 2.6.3.

Corollary 3.7.2. The Cartier divisor Y˚Pappmq on S˚Pap is Ok-flat, as is the

restriction of Z˚Krapmq to S˚Kra r Exc.

Proof. Fix a prime p Ă Ok. In general, if X is a Noetherian, proper, flatOk,p-scheme with geometrically normal fibers, one can find a finite unram-ified extension F kp such that every connected component of XOF has

geometrically irreducible fibers. This follows from [FGI`05, Proposition8.5.16].

Choose a finite extension F kp so that every connected component ofS˚PapOF has geometrically irreducible fibers. Proposition 3.6.1 implies that


ZΦpmq is Ok-flat for every proper cusp label representative Φ. Theorem3.7.1 now implies that Y˚Pappmq is Ok-flat when restricted to some etaleneighborhood U of the boundary of S˚PapOF .

The boundary, and hence the etale neighborhood U , meets every con-nected component of S˚PapF (this can be checked in the complex fiber, where

it follows from the fact that a k-hermitian space of signature pn´ 1, 1q withn ě 3 admits an isotropic k-line). It follows that U meets every connectedcomponent of S˚PapOF , and hence meets every irreducible component of the

special fiber. We deduce that the support of the Cartier divisor Y˚Pappmqcontains no irreducible components of the special fiber, so is Ok-flat.

As the isomorphism (3.7.2) identifies Y˚Pappmq with 2Z˚Krapmq, it follows

that the restriction of Z˚Krapmq to the complement of Exc is also flat.

3.8. Fourier-Jacobi expansions. We now define Fourier-Jacobi expan-sions of sections of the line bundle ωk of weight k modular forms on S˚Kra.

Fix a proper cusp label representative Φ “ pP, gq. Suppose ψ is a rationalfunction on S˚Kra, regular on an open neighborhood of the closed stratumS˚KrapΦq. Using the isomorphism (3.7.1) we obtain a formal function, againdenoted ψ, on the formal completion

pC˚Φq^BΦ“ SpfBΦ

´

ź

`ě0

L`Φ¯

.

Tautologically, there is a formal Fourier-Jacobi expansion

(3.8.1) ψ “ÿ

`ě0

FJ`pψq ¨ q`

with coefficients FJ`pψq P H0pBΦ,L`Φq.

Remark 3.8.1. Let π : C˚Φ Ñ BΦ be the natural map. The formal symbolq can be understood as follows: As C˚Φ is the total space of the line bundle

L´1Φ , there is a tautological section

q P H0pC˚Φ, π˚L´1Φ q

whose divisor is the zero section BΦ ãÑ C˚Φ. Any FJ` P H0pBΦ,L`Φq pulls

back to a section of π˚L`Φ, and so defines a function FJ` ¨ q` on C˚Φ.

In the same way, any rational section ψ of ωk on S˚Kra, regular on anopen neighborhood of S˚KrapΦq, admits a Fourier-Jacobi expansion (3.8.1),but now with coefficients

FJ`pψq P H0pBΦ,ω

kΦ b L`Φq – H0

`

E b L0,LiepEq´k bQ`EbL0

˘

.

Here E Ñ Mp1,0q is the universal CM elliptic curve, and we are using theisomorphism of Proposition 3.4.3 along with (3.5.1). By mild abuse of nota-tion, we have pulled back LiepEq´k via the natural map E b L0 ÑMp1,0q.


Consider the commutative diagram

ShpQΦ,DΦqpCq– //

CΦpCq // BΦpCq // AΦpCq

–

kˆzpkˆ pOˆk

aÞÑEpaq//Mp1,0qpCq.

Here the isomorphisms are those of Propositions 3.3.2 and 3.4.3, and thevertical arrow on the left is the surjection of Proposition 3.2.1. The bottomhorizontal arrow is defined as the unique function making the diagram com-mute. It is a bijection, and is given explicitly by the following recipe: Each

a P pkˆ determines a projective Ok-module

b “ a ¨HomOkpn, ga0q

of rank one. Identifying b with a fractional Ok-ideal, the elliptic curve Epaq

has complex points

(3.8.2) EpaqpCq “ Cb.

For each a P pkˆ there is a cartesian diagram

Epaq b L0//

E b L0

SpecpCq Epaq //Mp1,0q.

The Fourier-Jacobi coefficient

FJ`pψq P H0`

E b L0,LiepEq´k bQ`EbL0

˘

.

may be pulled back along the top map to yield a section

(3.8.3) FJpaq` pψq P H

0`

Epaq b L0,LiepEpaqq´k bQ`EpaqbL0

˘

.

3.9. Explicit coordinates. Once again, let Φ “ pP, gq be a proper cusplabel representative. We want to connect the algebraic theory of Fourier-Jacobi expansions of §3.8 to the classical analytic theory by choosing explicitcomplex coordinates, so as to identify (3.8.3) with a holomorphic functionon a complex vector space satisfying a particular transformation law.

The point of this discussion is to allow us, eventually, to show that theFourier-Jacobi coefficients of Borcherds products, expressed in the classicalway as holomorphic functions satisfying certain transformation laws, havealgebraic meaning. More precisely, the following discussion will be usedto deduce the algebraic statement of Proposition 6.4.1 from the analyticstatement of Proposition 6.3.1.

The explicit coordinates arise from a choice of Witt decomposition of thehermitian space V “ HomkpW0,W q, and the following lemma will allow usto choose this decomposition in a particularly nice way.


Lemma 3.9.1. The homomorphism νΦ of (3.1.1) admits a section

QΦ νΦ

// ReskQGm.

s

uu

This section may be chosen so that sp pOˆk q Ă KΦ, and such a choice deter-mines a decomposition

(3.9.1)ğ

aPkˆzpkˆ pOˆk

pQΦpQq X spaqKΦspaq´1qzDΦ – ShpQΦ,DΦqpCq,

where the isomorphism is z ÞÑ pz, spaqq on the copy of DΦ indexed by a.

Proof. Fix an isomorphism of hermitian Ok-modules

ga0 ‘ ga – ga0 ‘ gr´2pgaq ‘ gr´1pgaq ‘ gr0pgaq

as in Remark 3.1.3. After tensoring with Q, we let kˆ act on the right handside by a ÞÑ pa,Nmpaq, a, 1q. This defines a morphism kˆ Ñ GpQq, which,using (3.1.1), is easily seen to take values in the subgroup QΦpQq. Thisdefines the desired section s, and the decomposition (3.9.1) is immediatefrom Proposition 3.2.1.

Fix a section s as in Lemma 3.9.1. Recall from §3.1 the weight filtrationwtiV Ă V whose graded pieces

gr´1V “ HomkpW0, gr´2W q

gr0V “ HomkpW0, gr´1W q

gr1V “ HomkpW0, gr0W q

have k-dimensions 1, n ´ 2, and 1, respectively. Recalling (3.1.1), whichdescribes the action ofQΦ on the graded pieces of V , the section s determinesa splitting V “ V´1 ‘ V0 ‘ V1 of the weight filtration by

V´1 “ tv P V : @ a P kˆ, spaqv “ avu

V0 “ tv P V : @ a P kˆ, spaqv “ vu

V1 “ tv P V : @ a P kˆ, spaqv “ a´1vu.

The summands V´1 and V1 are isotropic k-lines, and V0 is the orthogonalcomplement of V´1 ` V1 with respect to the hermitian form on V . In par-ticular, the restriction of the hermitian form to V0 Ă V is positive definite.

Fix an a P pkˆ and define an Ok-lattice

L “ HomOkpspaqga0, spaqgaq Ă V.

Using the assumption sp pOˆk q Ă KΦ, we obtain a decomposition

L “ L´1 ‘ L0 ‘ L1


with Li “ L X Vi. The images of the lattices Li in the graded pieces griVare given by

L´1 “ a ¨HomOkpga0, gr´2pgaqq

L0 “ HomOkpga0, gr´1pgaqq

L1 “ a´1 ¨HomOkpga0, gr0pgaqq.

In particular, L0 is independent of a and agrees with (3.1.4).Choose a Z-basis e´1, f´1 P L´1, and let e1, f1 P δ

´1L1 be the dual basiswith respect to the (perfect) Z-bilinear pairing

r ¨ , ¨ s : L´1 ˆ δ´1L1 Ñ Z

obtained by restricting (2.1.6). This basis may be chosen so that

(3.9.2)L´1 “ Ze´1 ` Zf´1 δ´1L´1 “ Ze´1 `D

´1Zf´1

L1 “ Ze1 `DZf1 δ´1L1 “ Ze1 ` Zf1.

As εV1pCq Ă V1pCq is a line, there is a unique τ P C satisfying

(3.9.3) τe1 ` f1 P εV1pCq.

After possibly replacing both e1 and e´1 by their negatives, we may assumethat Impτq ą 0.

Proposition 3.9.2. The Z-lattice b “ Zτ ` Z is contained in k, and is afractional Ok-ideal. The elliptic curve

(3.9.4) EpaqpCq “ bzC

is isomorphic to (3.8.2), and there is an Ok-linear isomorphism of complexabelian varieties

(3.9.5) EpaqpCq b L0 – bL0zV0pRq.

Under this isomorphism the inverse of line bundle (3.4.2) has the form

(3.9.6) Q´1EpaqpCqbL0

– bL0zpV0pRq ˆ Cq,

where the action of y0 P bL0 on V0pRq ˆ C is

y0 ¨ pw0, qq “`

w0 ` εy0, q ¨ eπixy0,y0y

Npbq e´π

xw0,y0yImpτq

´πxy0,y0y2Impτq

˘

.

Proof. Consider the Q-linear map V´1 Ñ C defined by

αe´1 ` βf´1 ÞÑ ατ ` β.

It is k-linear, and identifies L´1 – b. It follows that b is Ok-stable. Using1 P b, we see that b Ă k, and is a fractional Ok-ideal. Moreover, we obtainan Ok-conjugate-linear isomorphism

(3.9.7) L´1αe´1`βf´1 ÞÑατ`βÝÝÝÝÝÝÝÝÝÝÝÝÑ b.


Exactly as in (2.1.4), the self-dual hermitian forms on ga0 and ga inducean Ok-conjugate-linear isomorphism

HomOkpga0, gr´2pgaqq – HomOk

pgr0pgaq, ga0q.

This determines an Ok-conjugate-linear isomorphism

L´1 – a ¨HomOkpgr0pgaq, ga0q “ a ¨HomOk

pn, ga0q,

which, when composed with (3.9.7), results in an Ok-linear isomorphism

b – a ¨HomOkpn, ga0q.

Thus the elliptic curves in (3.8.2) and (3.9.4) are isomorphic, and

EpaqpCq b L0 “ pbzCq b L0 – pbb L0qzpCb L0q.

Here, and throughout the remainder of the proof, all tensor products areover Ok. Identifying Cb L0 – V0pRq proves (3.9.5).

It remains to explain the isomorphism (3.9.6). First consider the Poincarebundle on the product

EpaqpCq ˆ EpaqpCq – pbˆ bqzpCˆ Cq.

Using classical formulas, the space of this line bundle can be identified withthe quotient

PEpaqpCq “ pbˆ bqzpCˆ Cˆ Cq,

where the action is given by

pb1, b2q ¨ pz1, z2, qq “´

z1 ` b1, z2 ` b2, q ¨ eπHτ pz1,b2q`πHτ pz2,b1q`πHτ pb1,b2q

¯

,

and we have set Hτ pw, zq “ wzImpτq for complex numbers w and z.Directly from the definition, the line bundle (3.4.2) on

EpaqpCq b L0 – pbb L0qzpCb L0q

is given by

QEpaqpCqbL0– pbb L0qz

`

pCb L0q ˆ C˘

,

where the action of bbL0 on pCbL0qˆC is given as follows: Choose any setx1, . . . , xn P L0 of Ok-module generators, and the extend the Ok-hermitianform on L0 to a C-hermitian form on Cb L0. If

y0 “ÿ

i

bi b xi P bb L0

and

w0 “ÿ

i

zi b xi P Cb L0

then

y0 ¨ pw0, qq “ pw0 ` y0, q ¨ eπX`πY q,


where the factors X and Y are

X “ÿ

iăj

´

Hτ pxxi, xjyzi, bjq `Hτ pzj , xxi, xjybiq `Hτ pxxi, xjybi, bjq¯

“1

Impτq

ÿ

i‰j

xzi b xi, bj b xjy `1

Impτq

ÿ

iăj

xbi b xi, bj b xjy

and, recalling γ “ p1` δq2,

Y “ÿ

i

´

Hτ pγxxi, xiyzi, biq `Hτ pzi, γxxi, xiybiq `Hτ pγxxi, xiybi, biq¯

“1

Impτq

ÿ

i

xzi b xi, bi b xiy `1

Impτq

ÿ

i

γxbi b xi, bi b xiy.

For elements y1, y2 P bb L0, we abbreviate

αpy1, y1q “xy1, y2y

δNpbq´xy2, y1y

δNpbqP Z.

Using 2iImpτq “ δNpbq, some elementary calculations show that

πX ` πY ´πxw0, y0y

Impτq

“2πi

δNpbq

ÿ

iăj

xbi b xi, bj b xjy `2πi

δNpbq

ÿ

i

xγbi b xi, bi b xiy

“π

2Impτq

ÿ

i,j

xbi b xi, bj b xjy ´πi

Npbq

ÿ

i,j

xbi b xi, bj b xjy

`2πiÿ

iăj

αpγbi b xi, bj b xjq `2πi

Npbq

ÿ

i

xbi b xi, bi b xiy.

All terms in the final line lie in 2πiZ, and so

eπX`πY “ eπxw0,y0y

Impτq eπxy0,y0y2Impτq e

´πixy0,y0y

Npbq .

The relation (3.9.6) follows immediately.

Proposition 3.9.2 allows us to express Fourier-Jacobi coefficients explicitlyas functions on V0pRq satisfying certain transformation laws: Suppose westart with a global section

(3.9.8) ψ P H0`

S˚KraC,ωk˘

.

For each a P pkˆ and ` ě 0 we have the algebraically defined Fourier-Jacobicoefficient

(3.9.9) FJpaq` pψq P H

0`

Epaq b L0,Q`EpaqbL0

˘

,

of (3.8.3), where we have trivialized the Lie algebra of Epaq using (3.9.4).The isomorphism (3.9.6) now identifies (3.9.9) with a function on V0pRq


satisfying the transformation law

(3.9.10) FJpaq` pψqpw0 ` y0q “ FJ

paq` pψqpw0q ¨ e

iπ`xy0,y0y

Npbq eπ`xw0,y0yImpτq

`π`xy0,y0y2Impτq

for all y0 P bL0.

Remark 3.9.3. If we use V0pRq – εV0pCq to view (3.9.9) as a function ofw0 P εV0pCq, the transformation law can be expressed in terms of the C-bilinear form as

FJpaq` pψqpw0 ` εy0q “ FJ

paq` pψqpw0q ¨ e

iπ`Qpy0qNpbq e

π`rw0,y0sImpτq

`π`Qpy0q2Impτq

for all y0 P bL0. This uses the (slightly confusing) commutativity of

V0pRq– //

x¨,x0y $$

εV0pCqĂ // V0pCq

r¨,x0szzC.

In order to give another interpretation of our explicit coordinates, letNΦ Ă QΦ be the unipotent radical, and let UΦ Ă NΦ be its center. Theunipotent radical may be characterized as the kernel of the morphism νΦ

of (3.1.1), or, equivalently, as the largest subgroup acting trivially on allgraded pieces griV .

Proposition 3.9.4. There is a commutative diagram

(3.9.11) pUΦpQq X spaqKΦspaq´1qzDΦ

z ÞÑpw0,qq //

εV0pCq ˆ Cˆ

pNΦpQq X spaqKΦspaq

´1qzDΦ// bL0zpεV0pCq ˆ Cˆq

in which the horizontal arrows are holomorphic isomorphisms, and the ac-tion of bL0 on

εV0pCq ˆ Cˆ – V0pRq ˆ Cˆ

is the same as in Proposition 3.9.2

Proof. Recall from Remark 3.2.3 the isomorphism

DΦ –

w P εV pCq : εV pCq “ εV´1pCq ‘ εV0pCq ‘ Cw(

Cˆ.

As εV pCq is totally isotropic with respect to r¨, ¨s, a simple calculation showsthat every line w P DΦ has a unique representative of the form

´ξpe´1 ´ τ f´1q ` w0 ` pτe1 ` f1q P εV´1pCq ‘ εV0pCq ‘ εV1pCq

with ξ P C and w0 P εV0pCq “ V0pRq. These coordinates define an isomor-phism of complex manifolds

(3.9.12) DΦw ÞÑpw0,ξqÝÝÝÝÝÝÑ εV0pCq ˆ C.


The action of G on V restricts to a faithful action of NΦ, allowing us toexpress elements of NΦpQq as matrices

npφ, φ˚, uq “

¨

˝

1 φ˚ u` 12φ˚ ˝ φ

1 φ1

˛

‚P NΦpQq

for maps

φ P HomkpV1, V0q, φ˚ P HomkpV0, V´1q, u P HomkpV1, V´1q

satisfying the relations

0 “ xφpx1q, y0y ` xx1, φ˚py0qy

0 “ xupx1q, y1y ` xx1, upy1qy

for xi, yi P Vi. The subgroup UΦpQq is defined by φ “ 0 “ φ˚.The group UΦpQqXspaqKΦspaq

´1 is cyclic, and generated by the elementnp0, 0, uq defined by

upx1q “xx1, ay

rL´1 : Okas¨ δa

for any a P L´1. In terms of the bilinear form, this can be rewritten as

upx1q “ ´rx1, f´1se´1 ` rx1, e´1sf´1.

In the coordinates of (3.9.12), the action of np0, 0, uq on DΦ becomes

pw0, ξq ÞÑ pw0, ξ ` 1q,

and setting q “ e2πiξ defines the top horizontal isomorphism in (3.9.11).Let V ´1 “ V´1 with its conjugate action of k. There are group isomor-

phisms

(3.9.13) NΦpQqUΦpQq – V ´1 bk V0 – V0.

The first sends

npφ, φ˚, uq ÞÑ y´1 b y0,

where y´1 and y0 are defined by the relation φpx1q “ xx1, y´1y ¨ y0, and thesecond sends (compare with (3.9.7))

pαe´1 ` βf´1q b y0 ÞÑ pατ ` βqy0.

A slightly tedious calculation shows that (3.9.13) identifies

pNΦpQq X spaqKΦspaq´1qpUΦpQq X spaqKΦspaq

´1q – bL0,

defining the bottom horizontal arrow in (3.9.11), and that the resultingaction of bL0 on εV0pCq ˆ Cˆ agrees with the one defined in Proposition3.9.2. We leave this to the reader.

Any section (3.9.8) may now be pulled back via

pNΦpQq X spaqKΦspaq´1qzD z ÞÑpz,spaqgq

ÝÝÝÝÝÝÝÑ ShpG,DqpCq


to define a holomorphic section of pωanqk, the kth power of the tautologicalbundle on

D –


Cˆ.The tautological bundle admits a natural NΦpRq-equivariant trivialization:any line w P D must satisfy rw, f´1s ‰ 0, yielding an isomorphism

r ¨ , f´1s : ωan – OD.

This trivialization allows us to identify ψ with a holomorphic function onD Ă DΦ, which then has an analytic Fourier-Jacobi expansion

(3.9.14) ψ “ÿ

`

FJpaq` pψqpw0q ¨ q

`

defined using the coordinates of Proposition 3.9.4. The fact that the coeffi-cients here agree with (3.9.9) is a special case of the main results of [Lan12],which compare algebraic and analytic Fourier-Jacobi coefficients on generalPEL-type Shimura varieties.

4. Classical modular forms

Throughout §4 we let D be any odd squarefree positive integer, and ab-breviate Γ “ Γ0pDq. Let k be any positive integer.

4.1. Weakly holomorphic forms. The positive divisors of D are bijectionwith the cusps of the complex modular curve X0pDqpCq, by sending r | Dto

8r “r

DP ΓzP1pQq.

Note that r “ 1 corresponds to the usual cusp at infinity, and so we some-times abbreviate 8 “ 81.

Fix a positive divisor r | D, set s “ Dr and choose

Rr “

ˆ

α βsγ rδ

˙

P Γ0psq

with α, β, γ, δ P Z. The corresponding Aktin-Lehner operator is defined bythe matrix

Wr “

ˆ

rα βDγ rδ

˙

“ Rr

ˆ

r1

˙

.

The matrix Wr normalizes Γ, and so acts on the cusps of X0pDqpCq. Thisaction satisfies Wr ¨ 8 “ 8r.

Let χ be a quadratic Dirichlet character modulo D, and let

χ “ χr ¨ χs

be the unique factorization as a product of quadratic Dirichlet charactersχr and χs modulo r and s, respectively. Write

MkpD,χq ĂM !kpD,χq


for the spaces of holomorphic modular forms and weakly holomorphic mod-ular forms of weight k, level Γ, and character χ. We assume that χp´1q “p´1qk, since otherwise M !

kpD,χq “ 0.

Denote by GL`2 pRq Ă GL2pRq the subgroup of elements with positivedeterminant. It acts on functions on the upper half plane by the usualweight k slash operator

pf |k γqpτq “ detpγqk2pcτ ` dq´kfpγτq, γ “

ˆ

a bc d

˙

P GL`2 pRq,

and f ÞÑ f |k Wr defines an endomorphism of M !kpD,χq satisfying

f |k W2r “ χrp´1qχsprq ¨ f.

In particular, Wr is an involution when χ is trivial.Any weakly holomorphic modular form

fpτq “ÿ

m"´8

cpmq ¨ qm PM !kpD,χq

determines another weakly holomorphic modular form

χrpβqχspαq ¨ pf |k Wrq PM!kpD,χq,

which is easily seen to be independent of the choice of parameters α, β, γ, δin the definition of Wr. This second modular form has a q-expansion at 8,denoted

(4.1.1) χrpβqχspαq ¨ pf |k Wrq “ÿ

m"´8

crpmq ¨ qm.

Definition 4.1.1. We call (4.1.1) the q-expansion of f at 8r. Of specialinterest is crp0q, the constant term of f at 8r.

Remark 4.1.2. If χ is nontrivial, the coefficients of (4.1.1) need not lie in thesubfield of C generated by the Fourier coefficients of f .

4.2. Eisenstein series and the modularity criterion. If k ą 2 we maydefine an Eisenstein series

E “ÿ

γPΓ8zΓ

χpdq ¨ p1 |k γq PMkpD,χq.

Here Γ8 Ă Γ is the stabilizer of 8 P P1pQq, and γ “`

a bc d

˘

P Γ.Define the (normalized) Eisenstein series for the cusp 8r by

Er “ χrp´βqχspαrq ¨ pE |k Wrq PMkpD,χq.

It is independent of the choice of the parameters in Wr, and we denote by

Erpτq “ÿ

mě0

erpmq ¨ qm

its q-expansion at 8.


Remark 4.2.1. Our notation for the q-expansion of Er is slightly at oddswith (4.1.1), as the q-expansion of E at 8r is not

ř

erpmqqm. Instead, the

q-expansion of E at 8r is χrp´1qχsprqř

erpmqqm, while the q-expansion of

Er at 8r isř

e1pmqqm. In any case, what matters most is that

constant term of Er at 8s “

#

1 if s “ r

0 otherwise.

Denote by

M !,82´kpD,χq ĂM !

2´kpD,χq

the subspace of weakly holomorphic forms that are holomorphic outside thecusp 8, and by

M8k pD,χq ĂMkpD,χq

the subspace of holomorphic modular forms that vanish at all cusps differentfrom 8. If k ą 2 then

M8k pD,χq “ CE ‘ SkpD,χq

for the weight k Eisenstein series E of §4.2. The constant terms of weakly

holomorphic modular forms in M !,82´kpD,χq can be computed using the above

Eisenstein series.

Proposition 4.2.2. Assume k ą 2. Suppose r | D and

fpτq “ÿ

m"´8

cpmq ¨ qm PM !,82´kpD,χq.

The constant term of f at the cusp 8r, in the sense of Definition 4.1.1,satisfies

crp0q `ÿ

mą0

cp´mqerpmq “ 0.

Proof. The meromorphic differential form fpτqErpτq dτ onX0pDqpCq is holo-morphic away from the cusps8 and8r. Summing its residues at these cuspsgives the desired equality.

Theorem 4.2.3 (Modularity criterion). Suppose k ě 2. For a formal powerseries

(4.2.1)ÿ

mě0

dpmqqm P Crrqss,

the following are equivalent:

(1) The relationř

mě0 cp´mqdpmq “ 0 holds for every weakly holomor-phic form

ÿ

m"´8

cpmq ¨ qm PM !,82´kpD,χq.

(2) The formal power series (4.2.1) is the q-expansion of a modular formin M8

k pD,χq.


Proof. As we assume k ě 2, that the map sending a weakly holomorphicmodular form f to its principal part at 8 identifies

M !,82´kpD,χq Ă Crq´1s.

On the other hand, the map sending a holomorphic modular form to itsq-expansion identifies

M8k pD,χq Ă Crrqss.

A slight variant of the modularity criterion of [Bor99, Theorem 3.1] showsthat each subspace is the exact annihilator of the other under the bilinearpairing Crq´1s b Crrqss ÝÑ C sending P b g to the constant term of P ¨ g.The claim follows.

5. Unitary Borcherds products

The goal of §5 is to state Theorems 5.3.1, 5.3.3, and 5.3.4, which assertsthe existence of Borcherds products on S˚Kra and S˚Pap having prescribeddivisors and prescribed leading Fourier-Jacobi coefficients. These theoremsare the technical core of this work, and their proofs will occupy all of §6.

5.1. Jacobi forms. In this section we recall some of the rudiments of thearithmetic theory of Jacobi forms. A more systematic treatment can befound in the work of Kramer [Kra91, Kra95].

Let Y be the moduli stack over Z classifying elliptic curves, and let π :E Ñ Y be the universal elliptic curve. Abbreviate Γ “ SL2pZq, and let H bethe complex upper half-plane. The groups Γ and Z2 each act on Hˆ C by

ˆ

a bc d

˙

¨ pτ, zq “

ˆ

aτ ` b

cτ ` d,

z

cτ ` d

˙

,

„

αβ

¨ pτ, zq “ pτ, z ` ατ ` βq ,

and this defines an action of the semi-direct product Γ˚ “ Γ ˙ Z2. Weidentify the commutative diagrams

(5.1.1) ΓzpHˆ Cq

%%

LiepEpCqq

exp

%%Γ˚zpHˆ Cq // ΓzH EpCq // YpCq

by sending pτ, zq P Hˆ C to the vector z in the Lie algebra of CpZτ ` Zq.Define a line bundle Opeq on E as the inverse ideal sheaf of the zero

section e : Y Ñ E . The Lie algebra LiepEq is (by definition) e˚Opeq, andωY “ LiepEq´1 is the usual line bundle of weight one modular forms on Y(see Remark 5.1.3 below). In particular, the line bundle

Q “ Opeq b π˚ωY

on E is canonically trivialized along the zero section. For a scheme U and apoint a P EpUq, denote by Qpaq the pullback of Q via a : U Ñ E .

Denote by P the pullback of the Poincare bundle via the canonical iso-morphism E ˆY E – E ˆY E_. If U is any scheme and a, b P EpUq, we


obtain a line bundle Ppa, bq on U exactly as in (3.4.1). There are canonicalisomorphisms

Ppa, bq – Qpa` bq bQpaq´1 bQpbq´1

and Ppa, aq – Qpaq bQpaq.

Definition 5.1.1. The diagonal restriction

J0,1 “ pdiagq˚P – Q2

is the line bundle of Jacobi forms of weight 0 and index 1 on E . Moregenerally,

Jk,m “ Jm0,1 b π

˚ωkYis the line bundle of Jacobi forms of weight k and index m on E .

The isomorphism of the following proposition is presumably well-known.We include the proof in order to make explicit the normalization of theisomorphism (see Remark 5.1.3 below, for example).

Proposition 5.1.2. Let p : H ˆ C Ñ EpCq be the quotient map. Theholomorphic line bundle J an

k,m on EpCq is isomorphic to the holomorphic

line bundle whose sections over an open set U Ă EpCq are holomorphicfunctions F pτ, zq on p´1pU q satisfying the transformation laws

F

ˆ

aτ ` b

cτ ` d,

z

cτ ` d

˙

“ F pτ, zq ¨ pcτ ` dqk ¨ e2πimcz2pcτ`dq

and

(5.1.2) F pτ, z ` ατ ` βq “ F pτ, zq ¨ e´2πimpα2τ`2αzq.

Proof. Let Jk,m be the holomorphic line bundle on EpCq defined by the abovetransformation laws.

By identifying the diagrams (5.1.1), a function f , defined on a Γ-invariantopen subset of H and satisfying the transformation law

f

ˆ

aτ ` b

cτ ` d

˙

“ fpτq ¨ pcτ ` dq´1

of a weight ´1 modular form, defines a section τ ÞÑ pτ, fpτqq of the linebundle

ΓzpHˆ Cq – LiepEpCqq – pωanY q´1

on ΓzH. This determines an isomorphism J1,0 – J an1,0 . It now suffices to

construct an isomorphism J0,1 – J an0,1 , and then take tensor products.

Fix τ P H, set Eτ “ CpZτ ` Zq, and restrict both J an0,1 and J0,1 to line

bundles on Eτ Ă EpCq. The imaginary part of the hermitian form

Hτ pz1, z2q “z1z2

Impτq

restricts to a Riemann form on Zτ`Z. By classical formulas for the Poincarebundle on complex abelian varieties, the restriction of J an

0,1 to the fiber Eτ


is isomorphic to the holomorphic line bundle determined by the Appell-Humbert data 2Hτ and the trivial character Zτ `ZÑ Cˆ. The sections ofthis holomorphic line bundle are, by definition, holomorphic functions gτ onC satisfying the transformation law

gτ pz ` `q “ gτ pzq ¨ e2πHτ pz,`q`πHτ p`,`q

for all ` P Zτ ` Z. If we set

F pτ, zq “ gτ pzq ¨ e´πHτ pz,zq,

this transformation law becomes (5.1.2).The above shows that J an

0,1 and J0,1 are isomorphic when restricted to the

fiber over any point τ P YpCq, but such an isomorphism is only determinedup to scaling by Cˆ. To pin down the scalars, and to get an isomorphism overthe total space, use the fact that both J an

0,1 and J0,1 come (by construction)with canonical trivializations along the zero section. By the Seesaw Theorem[BL04, Appendix A], there is a unique isomorphism J an

0,1 – J0,1 compatiblewith these trivializations.

Remark 5.1.3. The proof of Proposition 5.1.2 identifies a classical modularform fpτq “

ř

cpmqqm of weight k and level Γ with a holomorphic sectionof pωanY q

k, again denoted f , satisfying an additional growth condition at thecusp. Under our identification, the q-expansion principle takes the followingform: if R Ă C is any subring, then f is the analytification of a globalsection f P H0pYR,ωkYRq if and only if cpmq P p2πiqkR for all m.

For τ P H and z P C, we denote by

ϑ1pτ, zq “ÿ

nPZeπipn`

12q

2τ`2πipn` 1

2qpz´12q

the classical Jacobi theta function, and by

ηpτq “ eπiτ128ź

n“1

p1´ e2nπiτ q

Dedekind’s eta function. Set

Θpτ, zqdef“ i

ϑ1pτ, zq

ηpτq“ q112pζ12 ´ ζ´12q

8ź

n“1

p1´ ζqnqp1´ ζ´1qnq

where q “ e2πiτ and ζ “ e2πiz.

Proposition 5.1.4. The Jacobi form Θ24 defines a global section

Θ24 P H0pE ,J0,12q

with divisor 24e, while p2πiη2q12 determines a nowhere vanishing section

p2πiη2q12 P H0pY,ω12Y q.


Proof. It is a classical fact that p2πiη2q12 is a nowhere vanishing modularform of weight 12. Noting Remark 5.1.3, the q-expansion principle showsthat it descends to a section on YQ, and thus may be viewed as a rationalsection on Y. Another application of the q-expansion principle shows thatits divisor has no vertical components. Thus its divisor is trivial.

Classical formulas show that Θ24 defines a holomorphic section of J an0,12

with divisor 24e, and so the problem is to show that Θ24 is defined overQ, and extends to a section on the integral model with the stated divisors.One could presumably deduce this from the q-expansion principle for Jacobiforms as in [Kra91, Kra95]. We instead borrow an argument from [Sch98,§1.2], which requires only the more elementary q-expansion principle forfunctions on E .

Let d be any positive integer. The bilinear relations (3.4.1) imply that

the line bundle J d2

0,1 b rds˚J ´1

0,1 on E is canonically trivial, and so

θ24d “ Θ24d2

b rds˚Θ´24

defines a meromorphic function on EpCq. The crucial point is that θ24d

is actually a rational function defined over Q, and extends to a rationalfunction on the integral model E with divisor

(5.1.3) divpθ24d q “ 24

`

d2Er1s ´ Erds˘

.

As in [Sch98, p. 387], this follows by computing the divisor first in thecomplex fiber, then using the explicit formula

θ24d pτ, zq “ q2pd2´1qζ´12dpd´1q

˜

ź

ně0

p1´ qnζqd2

1´ qnζd

ź

ną0

p1´ qnζ´1qd2

1´ qnζ´d

¸24

and the q-expansion principle on E to see that the divisor has no verticalcomponents.

The line bundle ω12Y is trivial, and hence there are isomorphisms

J0,12 – Q24 – Opeq24 b π˚ω12Y – Opeq24.

Thus there is some Θ24 P H0pE ,J0,12q with divisor 24e, and the rationalfunction

θ24d “ Θ24d2

b rds˚Θ´24

on E also has divisor (5.1.3).

Consider the meromorphic function ρ “ Θ24Θ24 on EpCq. By comput-ing the divisor in the complex fiber, we see that ρ is a nowhere vanishingholomorphic function, and hence is constant. But this implies that

ρd2´1 “ θ24

d θ24d .

By what was said above, the right hand side is (the analytification of) a

nowhere vanishing function on E . This implies that ρd2´1 “ ˘1, and the

only way this can hold for all d ą 1 is if ρ “ ˘1.


Now consider the tower of modular curves

Y1pDq Ñ Y0pDq Ñ Yover SpecpZq parametrizing elliptic curves with Drinfeld Γ1pDq-level struc-ture, Γ0pDq-level structure, and no level structure, respectively. We denoteby E the universal elliptic curve over any one of these bases, and view theline bundle of Jacobi forms J0,12 as a line bundle on any one of the three uni-versal elliptic curves. Similarly, we view the Jacobi forms Θ24 and p2πiη2q12

of Proposition 5.1.4 as being defined over any one of these bases.The following lemma will be needed in §5.3.

Lemma 5.1.5. Let Q : Y1pDq Ñ E be the universal D-torsion point. Forany r | D the line bundle

(5.1.4)â

bPZDZb‰0rb“0

pbQq˚J0,12

on Y1pDq is canonically trivial, and its section

F 24r “

â

bPZDZb‰0rb“0

pbQq˚Θ24

admits a canonical descent, denoted the same way, to a section of the trivialbundle on Y0pDq.

Proof. If x1, . . . , xr are integers representing the r-torsion subgroup of ZDZ,then 6

ř

x2i ” 0 pmod Dq. The bilinear relations (3.4.1) therefore provide a

canonical isomorphismâ

bPZDZb‰0rb“0

PpbQ, bQqb12 –â

bPZDZb‰0rb“0

PpQ, 12b2Qq – PpQ, eq – OY1pDq

of line bundles on Y1pDq. This is the desired trivialization of (5.1.4). Thesection F 24

r is obviously invariant under the action of the diamond operatorson Y1pDq, and so descends to Y0pDq.

5.2. Borcherds’ quadratic identity. For the remainder of §5 we denoteby χk : pZDZqˆ Ñ t˘1u the Dirichlet character determined by the exten-sion kQ, abbreviate

(5.2.1) χ “ χn´2k ,

and fix a weakly holomorphic form

(5.2.2) fpτq “ÿ

m"´8

cpmqqm PM !,82´npD,χq

with cpmq P Z for all m ď 0.For a proper cusp label representative Φ as in Definition 3.1.1, recall the

self-dual hermitian Ok-lattice L0 of signature pn ´ 2, 0q defined by (3.1.4).


The hermitian form on L0 determines a quadratic form Qpxq “ xx, xy, withassociated Z-bilinear form rx1, x2s “ TrkQxx1, x2y of signature p2n´ 4, 0q.

The modularity criterion of Theorem 4.2.3 implies the following identityof quadratic forms on L0 b R.

Proposition 5.2.1 (Borcherds’ quadratic identity). For all u P L0 b R,

ÿ

xPL0

cp´Qpxqq ¨ ru, xs2 “ru, us

2n´ 4

ÿ

xPL0

cp´Qpxqq ¨ rx, xs.

Proof. The homogeneous polynomial

P pu, vq “ ru, vs2 ´ru, us ¨ rv, vs

2n´ 4

on L0 b R is harmonic in both variables u and v. For any fixed u P L0 b Rthere is a corresponding theta series

θpτ, u, P q “ÿ

xPL0

P pu, xq ¨ qQpxq P SnpD,χq.

The modularity criterion of Theorem 4.2.3 therefore shows that

ÿ

mą0

cp´mqÿ

xPL0Qpxq“m

ˆ

ru, xs2 ´ru, us ¨ rx, xs

2n´ 4

˙

“ 0

for all u P L0 b R. This implies the assertion.

Recall from (3.6.2) that every x P L0 determines a diagram

(5.2.3) BΦjx //

E

AΦ

j // Y0pDq,

where, changing notation slightly from §5.1, Y0pDq is now the open modularcurve over Ok. Recall also that BΦ carries a distinguished line bundle LΦ

defined by (3.3.1), used to define the Fourier-Jacobi expansions of (3.8.1).We will use Borcherds’ quadratic identity to relate the line bundle LΦ tothe line bundle J0,1 of Jacobi forms on E .

Proposition 5.2.2. The rational number

(5.2.4) multΦpfq “ÿ

mą0

m ¨ cp´mq

n´ 2¨#tx P L0 : Qpxq “ mu

lies in Z, and there is a canonical isomorphism

L2¨multΦpfqΦ –

â

mą0

â

xPL0Qpxq“m

j˚xJcp´mq0,1

of line bundles on BΦ.


Proof. Proposition 5.2.1 implies the equality of hermitian forms

ÿ

xPL0

cp´Qpxqq ¨ xu, xy ¨ xx, vy “xu, vy

2n´ 4

ÿ

xPL0

cp´Qpxqq ¨ rx, xs

“ xu, vy ¨multΦpfq

for all u, v P L0. As L0 is self-dual, we may choose u and v so that xu, vy “ 1,and the integrality of multΦpfq follows from the integrality of cp´mq.

Set E “ E ˆY0pDqAΦ, and use Proposition 3.4.3 to identify BΦ – EbL0.The pullback of the line bundle

â

mą0

â

xPL0Qpxq“m

j˚xJbcp´mq0,1 –

â

xPL0

j˚xJbcp´Qpxqq0,1

via any T -valued point a “ř

tib yi P EpT qbL0 is, in the notation of §3.4,

â

xPL0

PE´

ÿ

i

xyi, xyti,ÿ

j

xyj , xytj

¯bcp´Qpxqq

–â

i,j

â

xPL0

PE`

cp´Qpxqq ¨ xyi, xy ¨ xx, yjy ¨ ti, tj˘

–â

i,j

PE`

xyi, yjy ¨ ti, tj˘bmultΦpfq

– PEbL0pa, aqbmultΦpfq

– QEbL0paqb2¨multΦpfq.

This, along with the isomorphism QEbL0 – LΦ of Proposition 3.4.3, provesthat

L2¨multΦpfqΦ – Q2¨multΦpfq

EbL0–

â

mą0

â

xPL0Qpxq“m

j˚xJcp´mq0,1 .

5.3. The unitary Borcherds product. For a prime p dividing D define

(5.3.1) γp “ ε´np ¨ pD, pqnp ¨ invppVpq P t˘1,˘iu,

where invppVpq is the invariant of Vp “ HomkpW0,W q bQ Qp in the sense of(1.7.2), and

εp “

#

1 if p ” 1 pmod 4q

i if p ” 3 pmod 4q.

It is equal to the local Weil index of the Weil representation of SL2pZpq onSLp Ă SpVpq, where Vp is viewed as a quadratic space as in (2.1.6). This isexplained in more detail in §8.1. For any r dividing D we define

(5.3.2) γr “ź

p|r

γp.


Let crp0q denote the constant term of f at the cusp 8r, as in Definition4.1.1, and define

k “ÿ

r|D

γr ¨ crp0q.

We will see later in Corollary 6.1.4 that all γr ¨ crp0q P Q.For every m ą 0 define a divisor

(5.3.3) BKrapmq “m

n´ 2

ÿ

Φ

#tx P L0 : xx, xy “ mu ¨ S˚KrapΦq

with rational coefficients on S˚Kra. Here the sum is over all K-equivalenceclasses of proper cusp label representatives Φ in the sense of §3.2, L0 is thehermitian Ok-module of signature pn´ 2, 0q defined by (3.1.4), and S˚KrapΦqis the boundary divisor of Theorem 3.7.1. It follows immediately from thedefinition (5.2.4) that

ÿ

mą0

cp´mq ¨ BKrapmq “ÿ

Φ

multΦpfq ¨ S˚KrapΦq.

For m ą 0 define the total special divisor

ZtotKrapmq “ Z˚Krapmq ` BKrapmq,

where Z˚Krapmq is the special divisor defined on the open Shimura variety in§2.5, and extended to the toroidal compactification in Theorem 3.7.1.

The following theorems assert the existence of Borcherds products on S˚Kraand S˚Pap having prescribed divisors and prescribed leading Fourier-Jacobicoefficients. Their proofs will occupy all of §6.

Theorem 5.3.1. After possibly replacing the form f of (5.2.2) by a positiveinteger multiple, there is a rational section ψpfq of the line bundle ωk onS˚Kra with the following properties:

(1) In the generic fiber, the divisor of ψpfq is

divpψpfqqk “ÿ

mą0

cp´mq ¨ ZtotKrapmqk.

(2) For every proper cusp label representative Φ, the Fourier-Jacobi ex-pansion of ψpfq, in the sense of (3.8.1), along the boundary divisor

∆ΦzBΦ – S˚KrapΦq

has the form

ψpfq “ qmultΦpfqÿ

`ě0

ψ` ¨ q`,

where ψ` is a rational section of ωkΦ b LmultΦpfq``Φ over BΦ.

(3) For any Φ as above, the leading coefficient ψ0 admits a factorization

ψ0 “ p2πiη2Φqk b P horΦ b P vertΦ ,

where the three terms on the right are defined as follows: Directlyfrom the definition (3.5.1), there is an isomorphism ωΦ – j˚ωY ,


where j is the morphism of (5.2.3). This, along with Proposition5.1.4, allows us to define a nowhere vanishing section

p2πiη2Φqk “ j˚p2πiη2qk P H0pAΦ,ω

kΦq,

which we pull back to BΦ. Next, use the function

F 24r “

â

bPZDZb‰0rb“0

pbQq˚Θ24

on Y0pDq of Lemma 5.1.5 to define a rational function

P vertΦ “â

r|Drą1

j˚F γrcrp0qr

on AΦ, and again pull back to BΦ. Finally, using Proposition 5.2.2,define a rational section

P horΦ “â

mą0

â

xPL0xx,xy“m

j˚xΘcp´mq

of the line bundle LmultΦpfqΦ on BΦ.

These properties determine ψpfq uniquely.

Remark 5.3.2. In replacing f by a positive integer multiple, we are tacitlyassuming that the constants γrcrp0q and cp´mq are integer multiples of 24for all r | D and all m ą 0. This is necessary in order to guarantee k P Z,and to make sense of the three factors p2πiη2

Φqk, P horΦ , and P vertΦ .

In fact, we can strengthen Theorem 5.3.1 by computing precisely thedivisor of ψpfq on the integral model S˚Kra.

Theorem 5.3.3. The rational section ψpfq of ωk has divisor

divpψpfqq “ÿ

mą0


k

2¨ Exc`

ÿ

r|D

γrcrp0qÿ

p|r

S˚KraFp

´ÿ

mą0

cp´mq

2

ÿ

sPπ0pSingq

#tx P Ls : xx, xy “ mu ¨ Excs,

where p Ă Ok is the unique prime above p, Ls is the self-dual HermitianOk-lattice defined in §2.6, and Excs Ă Exc is the fiber over the components P π0pSingq.

It is possible to give a statement analogous to Theorem 5.3.3 for theintegral model S˚Pap. To do this we first define, exactly as in (5.3.3), aCartier divisor

YtotPappmq “ Y˚Pappmq ` 2BPappmq


with rational coefficients on S˚Pap. Here Y˚Pappmq is the Cartier divisor ofTheorem §3.7.1, and

BPappmq “m

n´ 2

ÿ

Φ

#tx P L0 : xx, xy “ mu ¨ S˚PappΦq.

It is clear from Theorem 3.7.1 that

(5.3.4) 2 ¨ ZtotKrapmq “ Ytot

Krapmq `ÿ

sPπ0pSingq


where YtotKrapmq denotes the pullback of Ytot

Pappmq via S˚Kra Ñ S˚Pap.

The isomorphism ΩKra – ω2 b OpExcq of Theorem 3.7.1 identifies theline bundles Ωk and ω2k in the generic fiber of S˚Kra. This allows us to view

ψpfq2 as a rational section of ΩkKra. As S˚Kra Ñ S˚Pap is an isomorphism in

the generic fiber, this section descends to a rational section of

(5.3.5) ΩkPap P PicpS˚Papq.

Theorem 5.3.4. When viewed as a rational section of (5.3.5), the Borcherdsproduct ψpfq2 has divisor

divpψpfq2q “ÿ

mą0

cp´mq ¨ YtotPappmq ` 2

ÿ

r|D

γrcrp0qÿ

p|r

S˚PapFp.

These three theorems will be proved simultaneously in §6. Briefly, wewill map our unitary Shimura variety ShpG,Dq to an orthogonal Shimuravariety, where a meromorphic Borcherds product is already known to exist.If we pull back this Borcherds product to ShpG,DqpCq, the leading coeffi-cient in its analytic Fourier-Jacobi expansion is known from [Kud16], up tomultiplication by some unknown constants.

By converting this analytic Fourier-Jacobi expansion into algebraic lan-guage, we will deduce the existence of a Borcherds product ψpfq satisfyingall of the properties stated in Theorem 5.3.1, up to some unknown constantsin the leading Fourier-Jacobi coefficient. These unknown constants are theκΦ’s appearing in Proposition 6.4.1. We can rescale the Borcherds productto make many κΦ “ 1 simultaneously.

After such a rescaling, the divisor of ψpfq2 on S˚Pap can be computed fromthe Fourier-Jacobi expansions. This proves Theorem 5.3.4, and Theorem5.3.3 follows by pulling back the divisor calculation via S˚Kra Ñ S˚Pap usingTheorem 2.6.3.

Using the divisor calculation of Theorem 5.3.4, we prove that all κΦ areroots of unity. Thus, after replacing f by a multiple, which replaces ψpfqby a power, we can force all κΦ “ 1. This proves Theorem 5.3.1.

5.4. A divisor calculation at the boundary. Let Φ be a proper cusplabel representative. The following proposition will be needed in the proofsof Theorems 5.3.1, 5.3.3, and 5.3.4.


Proposition 5.4.1. On BΦ we have the equality of divisors

divpP horΦ q “ÿ

mą0

cp´mqZΦpmq

divpP vertΦ q “ÿ

r|D

γrcrp0qÿ

p|r

BΦFp.

In particular, the divisors of P horΦ and P vertΦ are purely horizontal (Proposi-tion 3.6.1) and purely vertical, respectively.

Proof. Let E Ñ Y0pDq be the universal elliptic curve, and denote by e :Y0pDq Ñ E the 0-section. It is an effective Cartier divisor on E .

Directly from the definition of P horΦ we have the equality

divpP horΦ q “ÿ

mą0

cp´mq

24

ÿ

xPL0xx,xy“m

divpj˚xΘ24q.

Combining Proposition 5.1.4 with (3.6.1) shows thatÿ

xPL0xx,xy“m

divpj˚xΘ24q “ÿ

xPL0xx,xy“m

24j˚xpeq “ÿ

xPL0xx,xy“m

24ZΦpxq “ 24ZΦpmq,

and the first equality follows immediately.Recall the morphism j : AΦ Ñ Y0pDq of §3.6. For the second equality

it suffices to prove that the function F 24r on Y0pDq defined in Lemma 5.1.5

satisfies

(5.4.1) divpj˚F 24r q “ 24

ÿ

p|r

AΦFp.

Let C Ă E be the universal cyclic subgroup scheme of order D. For eachs | D denote by Crss Ă C the s-torsion subgroup, and by Crssˆ Ă Crss theclosed subscheme of generators. More precisely, noting that

Crss “ź

p|s

Crps,

we may use the Oort-Tate theory of group schemes of prime order (see[HR12] for a summary). Define

Crssˆ “ź

p|s

Crpsˆ,

where Crpsˆ denotes the closed subscheme of generators of Crps as in [HR12,§3.3]. Note that Crpsˆ coincides with the subscheme of points of exact orderp. See [HR12, Rem. 3.3.2].

There is an equality of Cartier divisors

1

24divpF 24

r q “`

Crrs ´ e˘

ˆE,e Y0pDq “ÿ

s|rs‰1

`

Crssˆ ˆE,e Y0pDq˘


on Y0pDq. Indeed, one can check this after pullback to Y1pDq, where it isclear from Proposition 5.1.4, which asserts that the divisor of the sectionΘ24 appearing in the definition of F 24

r is equal to 24e. If s is divisible bytwo distinct primes then

`

Crssˆ Ê,e Y0pDq˘

“ 0,

and hencedivpF 24

r q “ 24ÿ

p|r

`

Crpsˆ Ê,e Y0pDq˘

.

Now pull back this equality of Cartier divisors by j. Recall that j isdefined as the composition

AΦ –Mp1,0qiÝÑ Y0pDq,

where the isomorphism is the one provided by Proposition 3.4.3, and thearrow labeled i endows the universal CM elliptic curve E ÑMp1,0q with itscyclic subgroup scheme Erδs. Thus

(5.4.2) i˚divpF 24r q “ 24

ÿ

p|r

`

Erpsˆ Ê,e Mp1,0q

˘

,

where p denotes the unique prime ideal in Ok over p.

Fix a geometric point z : SpecpFalgp q Ñ Mp1,0q, and view z also as a

geometric point of E or E using

Mp1,0qeÝÑ E

iÝÑ E .

Let OE,z and OE,z denote the completed etale local rings of E and E at z.By Oort-Tate theory there is an isomorphism

OE,z –W rrX,Y, ZsspXY ´ wpq

for some uniformizer wp in the Witt ring W “ W pFalgp q. Compare with

[HR12, Theorem 3.3.1]. Under this isomorphism the 0-section of E is definedby the equation Z “ 0, and the divisor Crpsˆ is defined by Zp´1 ´X “ 0.Moreover, noting that the completed etale local ring of Mp1,0q at z can beidentified with OkbW , the natural map OE,z Ñ OE,z is identified with thequotient map

W rrX,Y, ZsspXY ´ wpq ÑW rrX,Y, ZsspXY ´ wp, X ´ uY q

for some u PWˆ.Under these identifications, the closed immersion

Erpsˆ Ê,e Mp1,0q ãÑMp1,0q

corresponds, on the level of completed local rings, to the quotient map

OMp1,0q,z W rrX,Y, ZsspXY ´ wp, X ´ uY, Zq

Falgp W rrX,Y, ZsspXY ´ wp, X ´ uY, Z, Z

p´1 ´Xq.


This implies that

Erpsˆ Ê,e Mp1,0q “Mp1,0qFalg

p.

The equality (5.4.1) is clear from this and (5.4.2).

6. Calculation of the Borcherds product divisor

In this section we prove Theorems 5.3.1, 5.3.3, and 5.3.4. Throughout §6we keep f as in (5.2.2), and again assume that cp´mq P Z for all m ě 0.

Recall that V “ HomkpW0,W q is endowed with the hermitian form xx, yyof (2.1.5), as well as the Q-bilinear form rx, ys of (2.1.6). The associatedquadratic form is

Qpxq “ xx, xy “rx, xs

2.

6.1. Vector-valued modular forms. Let L Ă V be any Ok-lattice, self-dual with respect to the hermitian form. The dual lattice of L with respectto the bilinear form r¨, ¨s is L1 “ δ´1L.

Let ω be the restriction to SL2pZq of the Weil representation of SL2ppQq(associated with the standard additive character of AQ) on the Schwartz-Bruhat functions on LbZ Af . The restriction of ω to SL2pZq preserves thesubspace SL “ CrL1Ls of Schwartz-Bruhat functions that are supported onpL1 and invariant under translations by pL. We obtain a representation

ωL : SL2pZq Ñ AutpSLq.

For µ P L1L, we denote by φµ P SL the characteristic function of µ.

Remark 6.1.1. The conjugate representation ωL on SL, defined by

ωLpγqpφq “ ωLpγqpφq

for φ P SL, is the representation denoted ρL in [Bor98, Bru02, BF04].

Recall the scalar valued modular form

fpτq “ÿ

m"´8

cpmq ¨ qm PM !,82ńpD,χq

of (5.2.2), and continue to assume that cpmq P Z for all m ď 0. We

will convert f into a CrL1Ls-valued modular form f , to be used as inputfor Borcherds’ construction of meromorphic modular forms on orthogonalShimura varieties. The restriction of ωL to Γ0pDq acts on the line C ¨ φ0 viathe character χ, and hence the induced function

fpτq “ÿ

γPΓ0pDqzSL2pZqpf |2ń γqpτq ¨ ωLpγq

´1φ0(6.1.1)

is an SL-valued weakly holomorphic modular form for SL2pZq of weight 2ńwith representation ωL. Its Fourier expansion is denoted

(6.1.2) fpτq “ÿ

m"´8

cpmq ¨ qm,


and we denote by cpm,µq the value of cpmq P SL at a coset µ P L1L.For any r | D let γr P t˘1,˘iu be as in (5.3.2), and let crpmq be the mth

Fourier coefficient of f at the cusp 8r as in (4.1.1). For any µ P L1L definerµ | D by

(6.1.3) rµ “ź

µp‰0

p,

where µp P L1pLp is the p-component of µ.

Proposition 6.1.2. For all m P Q the coefficients cpmq P SL satisfy

cpm,µq “

#

ř

rµ|r|Dγr ¨ crpmrq if m ” ´Qpµq pmod Zq,

0 otherwise.

Moreover, for m ă 0 we have

cpm,µq “

#

cpmq if µ “ 0,

0 if µ ‰ 0,

and the constant term of f is given by

cp0, µq “ÿ

rµ|r|D

γr ¨ crp0q.

Proof. The first formula is a special case of results of Scheithauer [Sch09,Section 5]. For the reader’s benefit we provide a direct proof in §8.2.

The formula for the m “ 0 coefficient is immediate from the generalformula. So is the formula for m ă 0, using the fact that the singularitiesof f are supported at the cusp at 8.

Remark 6.1.3. The first formula of Proposition 6.1.2 actually also holds forf in the larger space M !

2´npD,χq.

Corollary 6.1.4. The coefficients cpmq and cpmq satisfy the following:

(1) The cpmq are rational for all m.(2) The cpm,µq are rational for all m and µ, and are integral if m ă 0.(3) For all r | D we have γr ¨ crp0q P Q. In particular

cp0, 0q “ÿ

r|D

γr ¨ crp0q P Q.

Proof. For the first claim, fix any σ P AutpCQq. The form fσ ´ f P M !,82´n

is holomorphic at all cusps other than 8, and vanishes at the cusp 8 bythe assumption that as cpmq P Z for m ď 0. Hence fσ ´ f is a holomorphicmodular form of negative weight 2 ´ n, and therefore vanishes identically.It follows that cpmq P Q for all m.

Now consider the second claim. In view of the Proposition 6.1.2 the coef-ficients cpm,µq of f with m ă 0 are integers. Hence, for any σ P AutpCQq,the function fσ ´ f is a holomorphic modular form of weight 2 ´ n ă 0,which is therefore identically 0. Therefore f has rational Fourier coefficients.


The third claim follows from the second claim and the formula for theconstant term of f given in Proposition 6.1.2.

6.2. Construction of the Borcherds product. We now construct theBorcherds product ψpfq of Theorem 5.3.1 as the pullback of a Borcherdsproduct on the orthogonal Shimura variety defined by the quadratic spacepV,Qq. Useful references here include [Bor98, Bru02, Kud03, Hof14].

After Corollary 6.1.4 we may replace f by a positive integer multiple inorder to assume that cp´mq P 24Z for all m ě 0, and that γrcrp0q P 24Z forall r | D. In particular the rational number

k “ cp0, 0q

of Corollary 6.1.4 is an integer. Compare with Remark 5.3.2.Define a hermitian domain

(6.2.1) D “ tw P V pCq : rw,ws “ 0, rw,ws ă 0uCˆ.

Let ωan be the tautological bundle on D, whose fiber at w is the line Cw ĂV pCq. The group of real points of SOpV q acts on (6.2.1), and this actionlifts to an action on ωan.

As in Remark 2.1.3, any point z P D determines a line Cw Ă εV pCq. Thisconstruction defines a closed immersion

(6.2.2) D ãÑ D,

under which ωan pulls back to the line bundle ωan of §2.4. The hermitiandomain D has two connected components. Let D` Ă D be the connectedcomponent containing D.

For a fixed g P GpAq, we apply the constructions of §6.1 to the input formf and the self-dual hermitian Ok-lattice

L “ HomOkpga0, gaq Ă V.

The result is a vector-valued modular form f of weight 2´ n and represen-tation ωL : SL2pZq Ñ SL. The form f determines a Borcherds product Ψpfq

on D`; see [Bor98, Theorem 13.3] and Theorem 7.2.4. For us it is moreconvenient to use the rescaled Borcherds product

(6.2.3) ψgpfq “ p2πiqcp0,0qΨp2fq

determined by 2f . It is a meromorphic section of pωanqk.The subgroup SOpLq` Ă SOpLq of elements preserving the component

D` acts on ψgpfq through a finite order character [Bor00]. Replacing f by

mf has the effect of replacing ψgpfq by ψgpfqm, and so after replacing f by

a multiple we assume that ψgpfq is invariant under this action.

Denote by ψgpfq the pullback of ψgpfq via the map

pGpQq X gKg´1qzD Ñ SOpLq`zD`


induced by (6.2.2). It is a meromorphic section of pωanqk on the connectedcomponent

pGpQq X gKg´1qzD z ÞÑpz,gqÝÝÝÝÝÑ ShpG,DqpCq.

By repeating the construction for all g P GpQqzGpAf qK, we obtain a mero-

morphic section ψpfq of the line bundle pωanqk on all of ShpG,DqpCq.After rescaling on every connected component by a complex constant

of absolute value 1, this will be the section whose existence is asserted inTheorem 5.3.1.

6.3. Analytic Fourier-Jacobi coefficients. We return to the notation of§3.9. Thus Φ “ pP, gq is a proper cusp label representative, we have chosen

s : ReskQGm Ñ QΦ

as in Lemma 3.9.1, and have fixed a P pkˆ. This data determines a lattice

L “ HomOkpspaqga0, spaqgaq,

and Witt decompositions

V “ V´1 ‘ V0 ‘ V1, L “ L´1 ‘ L0 ‘ L1.

Choose bases e´1, f´1 P L´1 and e1, f1 P L1 as in §3.9.Imitating the construction of (3.9.12) yields a commutative diagram

Dp6.2.2q //

w ÞÑpw0,ξq

D`

w ÞÑpτ,w0,ξq

εV0pCq ˆ C // Hˆ V0pCq ˆ C

in which the vertical arrows are open immersions, and the horizontal arrowsare closed immersions. The vertical arrow on the right is defined as follows:Any w P D pairs nontrivially with the isotropic vector f´1, and so may bescaled so that rw, f´1s “ 1. This allows us to identify

D “ tw P V pCq : rw,ws “ 0, rw,ws ă 0, rw, f´1s “ 1u.

Using this model, any w P D` has the form

w “ ´ξe´1 ` pτξ ´Qpw0qqf´1 ` w0 ` τe1 ` f1

with τ P H, w0 P V0pCq, and ξ P C. The bottom horizontal arrow ispw0, ξq ÞÑ pτ, w0, ξq, where τ is determined by the relation (3.9.3).

The construction above singles out a nowhere vanishing section of ωan,whose value at an isotropic line Cw is the unique vector in that line withrw, f´1s “ 1. As in the discussion leading to (3.9.14), we obtain a trivializa-tion

r ¨ , f´1s : ωan – OD` .

Now consider the Borcherds product ψspaqgpfq on D` determined by thelattice L above (that is, replace g by spaqg throughout §6.2). It is a mero-morphic section of pωanqk, and we use the trivialization above to identify it


with a meromorphic function. In a neighborhood of the rational boundarycomponent associated to the isotropic plane V´1 Ă V , this meromorphicfunction has a product expansion.

Proposition 6.3.1 ([Kud16]). There are positive constants A and B with

the following property: For all points w P D` satisfying

Impξq ´QpImpw0qq

Impτqą A Impτq `

B

Impτq,

there is a factorization

ψspaqgpfq “ κ ¨ p2πiqk ¨ η2kpτq ¨ e2πiIξ ¨ P0pτq ¨ P1pτ, w0q ¨ P2pτ, w0, ξq

in which κ P Cˆ has absolute value 1, η is the Dedekind η-function, and

I “1

12

ÿ

bPZDZc

ˆ

0,´b

Df´1

˙

´ 2ÿ

mą0

ÿ

xPL0

cp´mq ¨ σ1pm´Qpxqq.

The factors P0 and P1 are defined by

P0pτq “ź

bPZDZb‰0

Θ

ˆ

τ,b

D

˙cp0, bD

f´1q

and

P1pτ, w0q “ź

mą0

ź

xPL0Qpxq“m

Θ`

τ, rw0, xs˘cp´mq

.

The remaining factor is

P2pτ, w0, ξq “ź

xPδ´1L0aPZ

bPZDZcPZą0

´

1´ e2πicξe2πiaτe2πibDe´2πirx,w0s¯2¨cpac´Qpxq,µq

,

where µ “ ´ae´1 ´bD f´1 ` x` ce1 P δ

´1LL.

Proof. This is just a restatement of [Kud16, Corollary 2.3], with some sim-

plifications arising from the fact that the vector-valued form f used to definethe Borcherds product is induced from a scalar-valued form via (6.1.1).

A more detailed description about how these expressions arise from thegeneral formulas in [Kud16] is given in the appendix.

If we pull back the formula for the Borcherds product ψspaqgpfq found inProposition 6.3.1 via the closed immersion (6.2.2), we obtain a formula forthe Borcherds product ψspaqgpfq on the connected component

pGpQq X spaqgKg´1spaq´1qzD z ÞÑpz,spaqgqÝÝÝÝÝÝÝÑ ShpG,DqpCq,

from which we can read off the leading analytic Fourier-Jacobi coefficient.


Corollary 6.3.2. The analytic Fourier-Jacobi expansion of ψpfq, in thesense of (3.9.14), has the form

ψspaqgpfq “ÿ

`ěI

FJpaq` pψpfqqpw0q ¨ q

`.

The leading coefficient FJpaqI pψpfqq, viewed as a function on V0pRq as in the

discussion leading to (3.9.10), is given by

(6.3.1) FJpaqI pψpfqqpw0q “ κ ¨ p2πiqk ¨ ηpτq2k ¨ P0pτq ¨ P1pτ, w0q,

where τ P H is determined by (3.9.3),

P0pτq “ź

r|D

ź

bPZDZb‰0rb“0

Θ

ˆ

τ,b

D

˙γrcrp0q

andP1pτ, w0q “

ź

mą0

ź

xPL0Qpxq“m

Θ`

τ, xw0, xy˘cp´mq

.

The constant κ P C, which depends on both Φ and a, has absolute value 1.

Proof. Using Proposition 6.3.1, the pullback of ψspaqgpfq via (6.2.2) factorsas a product

ψspaqgpfq “ κ ¨ p2πiqk ¨ η2kpτq ¨ e2πiξI ¨ P0pτqP1pτ, w0qP2pτ, w0, ξq,

where ξ P Cˆ and w0 P V pRq – εV pCq. The parameter τ P H is now fixed,determined by (3.9.3). The equality

ź

bPZDZb‰0

Θ

ˆ

τ,b

D

˙cp0, bD

f´1q

“ź

r|D

ź

bPZDZb‰0rb“0

Θ

ˆ

τ,b

D

˙γrcrp0q

follows from Proposition 6.1.2, and allows us to rewrite P0 in the stated form.To rewrite the factor P1 in terms of x¨, ¨y instead of r¨, ¨s, use the commutativediagram of Remark 3.9.3. Finally, as Impξq Ñ 8, so q “ e2πiξ Ñ 0, thefactor P2 converges to 1. This P2 does not contribute to the leading Fourier-Jacobi coefficient.

Proposition 6.3.3. The integer I defined in Proposition 6.3.1 is equal tothe integer multΦpfq defined by (5.2.4), and the product (6.3.1) satisfies thetransformation law (3.9.10) with ` “ multΦpfq.

Proof. The Fourier-Jacobi coefficient FJpaqI pψpfqq is, by definition, a section

of the line bundle QIEpaqbL

on Epaq b L. When viewed as a function of the

variable w0 P V0pRq using our explicit coordinates, it therefore satisfies thetransformation law (3.9.10) with ` “ I.

Now consider the right hand side of (6.3.1), and recall that τ is fixed, de-termined by (3.9.3). In our explicit coordinates the function Θpτ, xw0, xyq

24


of w0 P V0pRq is identified with a section of the line bundle j˚xJ0,12 on

Epaq b L; this is clear from the definition of jx in (3.6.2), and Proposition5.1.4. Thus P1pτ, w0q, and hence the entire right hand side of (6.3.1), definesa section of the line bundle

â

mą0

â

xPL0Qpxq“m

j˚xJcp´mq20,1 – L2¨multΦpf2q

Φ – QmultΦpfq

EpaqbL,

where the isomorphisms are those of Proposition 5.2.2 and Proposition 3.4.3.This implies that the right hand side of (6.3.1) satisfies the transformationlaw (3.9.10) with ` “ multΦpfq.

The claim now follows from (6.3.1) and the discussion above. For a moredirect proof of the proposition, see §8.4.

6.4. Algebraization and descent. The following weak form of Theorem5.3.1 shows that ψpfq is algebraic, and provides an algebraic interpretationof its leading Fourier-Jacobi coefficients.

Proposition 6.4.1. The meromorphic section ψpfq is the analytificationof a rational section of the line bundle ωk on SKraC. This rational sectionsatisfies the following properties:

(1) When viewed as a rational section over the toroidal compactification,

divpψpfqq “ÿ

mą0

cp´mq ¨ Z˚KrapmqC `ÿ

Φ

multΦpfq ¨ S˚KrapΦqC.

(2) For every proper cusp label representative Φ, the Fourier-Jacobi ex-pansion of ψpfq along S˚KrapΦqC, in the sense of §3.8, has the form

ψpfq “ qmultΦpfqÿ

`ě0

ψ` ¨ q`.

(3) The leading coefficient ψ0, a rational section of ωkΦ bLmultΦpfqΦ over

BΦC, factors as

ψ0 “ κΦ b p2πiη2Φqk b P horΦ b P vertΦ

for a unique section

κΦ P H0pAΦC,OˆAΦCq.

This section satisfies |κΦpzq| “ 1 at every complex point z P AΦpCq.(The other factors appearing on the right hand side were defined inTheorem 5.3.1.)

Proof. From Corollary 6.3.2 one can see that ψpfq extends to a meromorphicsection of ωk over the toroidal compactification S˚KrapCq, vanishing to orderI “ multΦpfq along the closed stratum S˚KrapΦqC Ă S˚KraC indexed by a

proper cusp label representative Φ. The calculation of the divisor of ψpfqover the open Shimura variety SKrapCq follows from the explicit calculation

of the divisor of ψpfq on the orthogonal Shimura variety due to Borcherds,


and the explicit complex uniformization of the divisors ZKrapmq found in[KR14] or [BHY15].

The algebraicity claim now follows from GAGA (using the fact that thedivisor is already known to be algebraic), proving all parts of the first claim.The second and third claims are just a translation of Corollary 6.3.2 intothe algebraic language of Theorem 5.3.1, using the explicit coordinates of§3.9.

We now prove that ψpfq, after minor rescaling, descends to k.Recall from Proposition 2.1.1 that the geometric components of ShpG,Dq

are defined over the Hilbert class field H Ă C of k, and that each suchcomponent has trivial stabilizer in GalpHkq. This allows us to chooseconnected components Xi Ă ShpG,DqH in such a way that

ShpG,DqH “ğ

i

ğ

σPGalpHkq

σpXiq.

For each index i, pick gi P GpAf q in such a way that XipCq is equal tothe image of

pGpQq X giKg´1i qzD

z ÞÑpz,giqÝÝÝÝÝÑ ShpG,DqpCq.

Choose an isotropic k-line J Ă W , let P Ă G be its stabilizer, and define aproper cusp label representative Φi “ pP, giq. The above choices pick out oneboundary component on every component of the toroidal compactification,as the following lemma demonstrates.

Lemma 6.4.2. The natural mapsŮ

i S˚KrapΦiq //

–

S˚Kra

Ů

iAΦi

Ů

i BΦi

88

&&

oo

Ů

i S˚PappΦiq // S˚Pap

induce bijections on connected components. The same is true after basechange to k or C.

Proof. Let X˚i Ă S˚PappCq be the closure of Xi. By examining the com-

plex analytic construction of the toroidal compactification [How15, Lan12,Pin89], one sees that some component of the divisor S˚PappΦiqpCq lies on

X˚i . Now recall from Theorem 3.7.1 that the components of S˚PappΦiqpCqare defined over H, and that the action of GalpHkq is simply transitive. Itfollows immediately that

S˚PappΦiqpCq Ăğ

σPGalpHkq

σpX˚i q,


and the inclusion induces a bijection on components. By Proposition 3.2.1and the isomorphism of Proposition 3.3.2, the quotient map

CΦpCq Ñ ∆ΦizCΦipCq

induces a bijection on connected components, and both maps CΦ Ñ BΦ Ñ

AΦ have geometrically connected fibers (the first is a Gm-torsor, and thesecond is an abelian scheme). We deduce that all maps in

AΦipCq Ð BΦipCq Ñ ∆ΦizBΦipCq – S˚KrapΦiqpCq – S˚PappΦiqpCq

induce bijections on connected components.The above proves the claim over C, and the claim over k follows formally

from this. The claim for integral models follows from the claim in the genericfiber, using the fact that all integral models in question are normal and flatover Ok.

Proposition 6.4.3. After possibly rescaling by a constant of absolute value1 on every connected component of S˚KraC, the Borcherds product ψpfq is

defined over k, and the sections of Proposition 6.4.1 satisfy

κΦ P H0pAΦk,OˆAΦk

q

for all proper cusp label representatives Φ. Furthermore, we may arrangethat κΦi “ 1 for all i.

Proof. Lemma 6.4.2 establishes a bijection between the connected compo-nents of S˚KrapCq and the finite set

Ů

iAΦipCq. On the component indexedby z P AΦipCq, rescale ψpfq by κΦipzq

´1. For this rescaled ψpfq we haveκΦi “ 1 for all i.

Suppose σ P AutpCkq. The second claim of Proposition 6.4.1 impliesthat the divisor of ψpfq, when computed on the compactification S˚KraC,

is defined over k. Therefore σpψpfqqψpfq has trivial divisor, and so isconstant on every connected component.

By Proposition 6.4.1, the leading coefficient in the Fourier-Jacobi expan-sion of ψpfq along the boundary stratum S˚KrapΦiq is

ψ0 “ p2πiη2Φiq

k b P horΦi b PvertΦi ,

which is defined over k. From this it follows that σpψpfqqψpfq is identicallyequal to 1 on every connected component of S˚KraC meeting this boundary

stratum. Varying i and using Lemma 6.4.2 shows that σpψpfqq “ ψpfq.This proves that ψpfq is defined over k, hence so are all of its Fourier-

Jacobi coefficients along all boundary strata S˚KrapΦq. Appealing again tothe calculation of the leading Fourier-Jacobi coefficient of Proposition 6.4.1,we deduce finally that κΦ is defined over k for all Φ.

6.5. Calculation of the divisor, and completion of the proof. TheBorcherds product ψpfq on S˚Krak of Proposition 6.4.3 may now be viewed

as a rational section of ωk on the integral model S˚Kra.


Let Φ be any proper cusp label representative. Combining Propositions6.4.1 and 6.4.3 shows that the leading Fourier-Jacobi coefficient of ψpfqalong the boundary divisor S˚KrapΦq is

(6.5.1) ψ0 “ κΦ b p2πiη2Φqk b P horΦ b P vertΦ .

Recall that this is a rational section of ωkΦ b LmultΦpfqΦ on BΦ. Here, by

mild abuse of notation, we are viewing κΦ as a rational function on AΦ, anddenoting in the same way its pullback to any step in the tower

C˚ΦπÝÑ BΦ Ñ AΦ.

Lemma 6.5.1. Recall that π has a canonical section BΦ ãÑ C˚Φ, realizingBΦ as a divisor on C˚Φ. If we use the isomorphism (3.7.1) to view ψpfq as a

rational section of the line bundle ωkΦ on the formal completion pC˚Φq^BΦ, its

divisor satisfies

divpψpfqq “ divpκΦq `multΦpfq ¨ BΦ

`ÿ

mą0

cp´mqZΦpmq `ÿ

r|D

γrcrp0qÿ

p|r

π˚pBΦFpq.

Proof. First we prove

(6.5.2) divpψpfqq “ π˚divpψ0q `multΦpfq ¨ BΦ.

Recalling the tautological section q with divisor BΦ from Remark 3.8.1,consider the rational section

R “ q´multΦpfq ¨ψpfq “ÿ

iě0

ψi ¨ qi

of ωkΦ b π˚LmultΦpfq

Φ on the formal completion pC˚Φq^BΦ.

We claim that divpRq “ π˚∆ for some divisor ∆ on BΦ. Indeed, whateverdivpRq is, it may decomposed as a sum of horizontal and vertical compo-nents. We know from Theorem 3.7.1 and Proposition 6.4.1 that the hori-zontal part is a linear combination of the divisors ZΦpmq on C˚Φ defined by(3.6.1); these divisors are, by definition, pullbacks of divisors on BΦ. On theother hand, the morphism C˚Φ Ñ BΦ is the total space of a line bundle, andhence is smooth with connected fibers. Thus every vertical divisor on C˚Φ,and in particular the vertical part of divpRq, is the pullback of some divisoron BΦ.

Denoting by i : BΦ ãÑ C˚Φ the zero section, we compute

∆ “ i˚π˚∆ “ i˚divpRq “ divpi˚Rq “ divpψ0q.

Pulling back by π proves that divpRq “ π˚divpψ0q, and (6.5.2) follows.We now compute the divisor of ψ0 on BΦ using (6.5.1). The factor

p2πiη2Φqk is nowhere vanishing by Proposition 5.1.4, and so does not con-

tribute to the divisor. The divisors of P horΦ and P vertΦ were computed in


Proposition 5.4.1, which shows that on BΦ we have the equality

divpψ0q “ divpκΦq `ÿ

mą0

cp´mqZΦpmq `ÿ

r|D

γrcrp0qÿ

p|r

BΦFp.

Combining this with (6.5.2) completes the proof.

Proposition 6.5.2. When viewed as a rational section of ωk on S˚Kra, theBorcherds product ψpfq has divisor

divpψpfqq “ÿ

mą0

cp´mq ¨ Z˚Krapmq `ÿ

Φ

multΦpfq ¨ S˚KrapΦq

`ÿ

r|D

γrcrp0qÿ

p|r

S˚KraFp(6.5.3)

up to a linear combination of irreducible components of the exceptional di-visor Exc Ă S˚Kra. Moreover, the sections κΦ of Proposition 6.4.3 are Z-torsion, and extend to sections κΦ P H

0pAΦ,OˆAΦq.

Proof. Arguing as in the proof of Corollary 3.7.2, we may find a finite ex-tension F kp such that the maps

Ů

i BΦiOF//

Ů

i S˚PappΦqOF// S˚PapOF

Ů

iAΦiOF

of Lemma 6.4.2 induce bijections on connected components, as well as onconnected (=irreducible) components of the generic and special fibers. Thesame property then holds if we replace S˚Pap by its dense open substack

X def“ S˚Pap r Sing – S˚Kra r Exc.

Suppose Φ is any proper cusp label representative, and let UΦ Ă XOF bethe union of all connected components that meet S˚PappΦqOF . If we interpret

divpκΦqOF as a divisor on XOF using the isomorphism

tvertical divisors on AΦOF u – tvertical divisors on XOF u,

then the equality of divisors (6.5.3) holds after pullback to UΦ, up to theerror term divpκΦqOF . Indeed, this equality holds in the generic fiber of UΦ

by Proposition 6.4.1, and it holds over an open neighborhood of S˚PappΦqOFby Lemma 6.5.1 and the isomorphism of formal completions (3.7.1). As theunion of the generic fiber with this open neighborhood is an open substackwhose complement has codimension ě 2, the stated equality holds over allof UΦ.

Letting Φ vary over the Φi and using κΦi “ 1, we see from the paragraphabove that (6.5.3) holds after base change to

Ů

i UΦi “ XOF . Using faithfullyflat descent for divisors and allowing p to vary proves (6.5.3). Now thatthe equality (6.5.3) is known, we may reverse the argument of the previous


paragraph to see that the error term divpκΦq vanishes for every Φ. It followsthat κΦ extends to a global section of OˆAΦ

.It only remains to show that each κΦ is a torsion section. Choose a finite

extension Lk large enough that every elliptic curve over C with complexmultiplication by Ok admits a model over L with everywhere good reduction.Choosing such models determines a faithfully flat morphism

ğ

SpecpOLq ÑMp1,0q – AΦ,

and the pullback of κΦ is represented by a tuple of units px`q Pś

OˆL . Eachx` has absolute value 1 at every complex embedding of L (this follows fromthe final claim of Proposition 6.4.1), and is therefore a root of unity. Thisimplies that κΦ is a torsion section.

Proof of Theorem 5.3.1. Start with a weakly holomorphic form (5.2.2). Asin §6.2, after possibly replacing f by a positive integer multiple, we obtaina Borcherds product ψpfq. This is a meromorphic section of pωanqk. ByProposition 6.4.1 it is algebraic, and by Proposition 6.4.3 it may be rescaledby a constant of absolute value 1 on each connected component in such away that it descends to k.

Now view ψpfq as a rational section of ωk over S˚Kra. By Proposition6.5.2 we may replace f by a further positive integer multiple, and replaceψpfq by a corresponding tensor power, in order to make all κΦ “ 1. Havingtrivialized the κΦ, the existence part of Theorem 5.3.1 now follows fromProposition 6.4.1. For uniqueness, suppose ψ1pfq also satisfies the conditionsof that theorem. The quotient of the two Borcherds products is a rationalfunction with trivial divisor, which is therefore constant on every connectedcomponent of S˚KrapCq. As the leading Fourier-Jacobi coefficients of ψ1pfqand ψpfq are equal along every boundary stratum, those constants are allequal to 1.

Proof of Theorem 5.3.4. As in the statement of the theorem, we now viewψpfq2 as a rational section of the line bundle Ωk

Pap on S˚Pap. CombiningProposition 6.5.2 with the isomorphism

S˚Kra r Exc – S˚Pap r Sing,

of (3.7.2), and recalling from Theorem 3.7.1 that this isomorphism identifies

ω2k – ΩkKra – Ωk

Pap,

we deduce the equality

divpψpfq2q “ÿ

mą0

cp´mq ¨ Y˚Pappmq ` 2ÿ

Φ

multΦpfq ¨ S˚PappΦq

` 2ÿ

r|D

γrcrp0qÿ

p|r

S˚PapFp(6.5.4)

of Cartier divisors on S˚Pap r Sing. As S˚Pap is normal and Sing lies incodimension ě 2, this same equality must hold on the entirety of S˚Pap.


Proof of Theorem 5.3.3. Now pull back (6.5.4) via S˚Kra Ñ S˚Pap. If we view

ψpfq2 as a rational section of the line bundle

ΩkKra – ω

2k bOpExcqk,

the equality (5.3.4) implies

divpψpfq2q “ÿ

mą0

cp´mq ¨ Y˚Krapmq ` 2ÿ

r|D

γrcrp0qÿ

p|r

S˚KraFp

` 2ÿ

Φ

multΦpfq ¨ S˚KrapΦq

“ 2ÿ

mą0

cp´mq ¨ Z˚Krapmq ` 2ÿ

r|D

γrcrp0qÿ

p|r

S˚KraFp

´ÿ

mą0

cp´mqÿ

sPπ0pSingq


` 2ÿ

Φ

multΦpfq ¨ S˚KrapΦq.

If we instead view ψpfq2 as a rational section of ω2k, this becomes

divpψpfq2q “ 2ÿ

mą0

cp´mq ¨ Z˚Krapmq ` 2ÿ

r|D

γrcrp0qÿ

p|r

S˚KraFp

´ÿ

mą0

cp´mqÿ

sPπ0pSingq


` 2ÿ

Φ

multΦpfq ¨ S˚KrapΦq ´ k ¨ Exc,

as desired.

7. Modularity of the generating series

Now armed with the modularity criterion of Theorem 4.2.3 and the arith-metic theory of Borcherds products on provided by Theorems 5.3.1, 5.3.3,and 5.3.4, we prove our main results: the modularity of generating series ofdivisors on the integral models S˚Kra and S˚Pap of the unitary Shimura variety

ShpG,Dq.The strategy follows that of [Bor99], which proves modularity of the gen-

erating series of divisors on the complex fiber of an orthogonal Shimuravariety.

7.1. The modularity theorems. Denote by

Ch1QpS˚Kraq – PicpS˚Kraq bZ Q

the Chow group of rational equivalence classes of Cartier divisors on S˚Krawith Q coefficients, and similarly for S˚Pap. There is a natural pullback map

Ch1QpS˚Papq Ñ Ch1

QpS˚Kraq.


Definition 7.1.1. Let χ be the quadratic Dirichlet character (5.2.1). If Vis any Q-vector space, a formal q-expansion

(7.1.1) g “ÿ

mě0

dpmq ¨ qm P V rrqss

is said to modular of level D, weight n, and character χ if for any linear mapρ : V Ñ C the q-expansion

ρpgq “ÿ

mě0

ρpdpmqq ¨ qm P Crrqss

is the q-expansion of an element of MnpD,χq.

Remark 7.1.2. If (7.1.1) is modular then its coefficients dpmq span a subspaceof V of dimension ď dimMnpD,χq. We leave the proof as an exercise forthe reader.

We also define the notion of the constant term of (7.1.1) at a cusp 8r,generalizing Definition 4.1.1.

Definition 7.1.3. Suppose a formal q-expansion g P V rrqss is modular oflevel D, weight n, and character χ. For any r | D, a vector v P V pCq is saidto be the constant term of g at the cusp 8r if, for every linear functionalρ : V pCq Ñ C, ρpvq is the constant term of ρpgq at the cusp 8r in the senseof Definition 4.1.1.

For m ą 0 we have defined in §5.3 effective Cartier divisors

YtotPappmq ãÑ S˚Pap, Ztot

Krapmq ãÑ S˚Kra

related by (5.3.4). We have defined in §3.7 line bundles

ΩPap P PicpS˚Papq, ω P PicpS˚Kraq

extending the line bundles on the open integral models defined in §2.4. Fornotational uniformity, we define

YtotPapp0q “ Ω´1

Pap, ZtotKrap0q “ ω

´1.

Theorem 7.1.4. The formal q-expansionÿ

mě0

YtotPappmq ¨ q

m P Ch1QpS˚Papqrrqss,

is a modular form of level D, weight n, and character χ. For any r | D, itsconstant term at the cusp 8r is

γr ¨´

YtotPapp0q ` 2

ÿ

p|r

S˚PapFp

¯

P Ch1QpS˚Papq bQ C.

Here γr P t˘1,˘iu is defined by (5.3.2), p Ă Ok is the unique prime abovep | r, and Fp is its residue field.


Proof. Let f be a weakly holomorphic form as in (5.2.2), and assume again

that cpmq P Z for all m ď 0. The space M !,82´npD,χq is spanned by such

forms. The Borcherds product ψpfq of Theorem 5.3.1 is a rational sectionof the line bundle

ωk “â

r|D

ωγrcrp0q,

on S˚Kra. If we view ψpfq2 as a rational section of the line bundle

ΩkPap –

â

r|D

Ωγrcrp0qPap

on S˚Pap, exactly as in Theorem 5.3.4, then

divpψpfq2q “ÿ

r|D

γrcrp0q ¨ YtotPapp0q

holds in the Chow group of S˚Pap. Comparing this with the calculation of

the divisor of ψpfq2 found in Theorem 5.3.4 shows that

0 “ÿ

mě0

cp´mq ¨ YtotPappmq `

ÿ

r|Drą1

γrcrp0q ¨ pYtotPapp0q ` 2Vrq,

where we abbreviate Vr “ř

p|r S˚PapFp.

For each r | D we have defined in §4.2 an Eisenstein series

Erpτq “ÿ

mě0

erpmq ¨ qm PMnpD,χq,

and Proposition 4.2.2 allows us to rewrite the above equality as

0 “ÿ

mě0

cp´mq ¨”

YtotPappmq ´

ÿ

r|Drą1

γrerpmq ¨ pYtotPapp0q ` 2Vrq

ı

.

Note that we have used erp0q “ 0 for r ą 1, a consequence of Remark 4.2.1.The modularity criterion of Theorem 4.2.3 now shows that

ÿ

mě0

YtotPappmq ¨ q

m ´ÿ

r|Drą1

γrEr ¨ pYtotPapp0q ` 2Vrq

is a modular form of level D, weight n, and character χ, whose constantterm vanishes at every cusp different from 8.

The theorem now follows from the modularity of each Er, together withthe description of their constant terms found in Remark 4.2.1.

Theorem 7.1.5. The formal q-expansionÿ

mě0

ZtotKrapmq ¨ q

m P Ch1QpS˚Kraqrrqss,

is a modular form of level D, weight n, and character χ.


Proof. The proof is essentially the same that of Theorem 7.1.4. One onlyneeds to replace the divisor calculation of Theorem 5.3.4 with the divisorcalculation of Theorem 5.3.3.

Indeed, by that theorem the rational section ψpfq of ωk has divisor

divpψpfqq “ÿ

mą0


k

2¨ Exc`

ÿ

r|D

γrcrp0qÿ

p|r

S˚KraFp

´ÿ

mą0

cp´mq

2

ÿ

sPπ0pSingq

#tx P Ls : xx, xy “ mu ¨ Excs.

On the other hand, the isomorphism ωk –Â

r|D ωγrcrp0q of line bundles

implies the relation

divpψpfqq “ÿ

r|D

γrcrp0q ¨ ZtotKrap0q

in the Chow group of S˚Kra.Combining these two equalities shows that

0 “ÿ

mě0

cp´mq ¨

¨

˝ZtotKrapmq ´

ÿ

sPπ0pSingq


2¨ Excs

˛

‚

`ÿ

r|Drą1

γrcrp0q

¨

˝´Exc

2` Ztot

Krap0q `ÿ

p|r

S˚KraFp

˛

‚

in Ch1QpS˚Kraq. Using Proposition 4.2.2 we rewrite this as

0 “ÿ

mě0

cp´mq ¨

¨

˝ZtotKrapmq ´

ÿ

sPπ0pSingq


2¨ Excs

˛

‚

`ÿ

mě0

cp´mqÿ

r|Drą1

γrerpmq

¨

˝

Exc

2´ Ztot

Krap0q ´ÿ

p|r

S˚KraFp

˛

‚,

where we have again used the fact that erp0q “ 0 for r ą 1.The modularity criterion of Theorem 4.2.3 now implies the modularity of

ÿ

mě0

ZtotKrapmq ¨ q

m ´1

2

ÿ

sPπ0pSingq

ϑspτq ¨ Excs

`ÿ

r|Drą1

γrErpτq ¨

¨

˝

Exc

2´ Ztot

Krap0q ´ÿ

p|r

S˚KraFp

˛

‚.


The theorem follows from the modularity of the Eisenstein series Erpτq andthe theta series

ϑspτq “ÿ

xPLs

qxx,xy PMnpD,χq.

7.2. Green functions. Here we construct Green functions for special divi-sors on S˚Kra as regularized theta lifts of harmonic Maass forms.

Recall from Section 2 the isomorphism of complex orbifolds

SKrapCq – ShpG,DqpCq “ GpQqzD ˆGpAf qK.

We use the uniformization on the right hand side and the regularized thetalift to construct Green functions for the special divisors

ZtotKrapmq “ Z˚Krapmq ` BKrapmq

on S˚Kra. The construction is a variant of the ones in [BF04] and [BHY15]adapted to our situation.

We now recall some of the basic notions of the theory of harmonic Maassforms, as in [BF04, Section 3]. Let H82ńpD,χq denote the space of harmonicMaass forms f of weight 2´ n for Γ0pDq with character χ such that

‚ f is bounded at all cusps of Γ0pDq different from the cusp 8,‚ f has polynomial growth at 8, in sense that there is a

Pf “ÿ

mă0

c`pmqqm P Crq´1s

such that f ´ Pf is bounded as q goes to 0.

A harmonic Maass form f P H82ńpD,χq has a Fourier expansion of the form

fpτq “ÿ

mPZm"´8

c`pmqqm `ÿ

mPZmă0

c´pmq ¨ Γ`

n´ 1, 4π|m| Impτq˘

¨ qm,(7.2.1)

where

Γps, xq “

ż 8

xe´tts´1dt

is the incomplete gamma function. The first summand on the right handside of (7.2.1) is denoted by f` and is called the holomorphic part of f , thesecond summand is denoted by f´ and is called the non-holomorphic part.

If f P H82ńpD,χq then (6.1.1) defines an SL-valued harmonic Maassform for SL2pZq of weight 2 ´ n with representation ωL. Proposition 6.1.2extends to such lifts of harmonic Maass forms, giving the same formulas forthe coefficients c`pm,µq of the holomorphic part f` of f . In particular, ifm ă 0 we have

c`pm,µq “

#

c`pmq if µ “ 0,

0 if µ ‰ 0,(7.2.2)


and the constant term of f is given by

c`p0, µq “ÿ

rµ|r|D

γr ¨ c`r p0q.

The formula of Proposition 4.2.2 for the contant terms c`r p0q of f at theother cusps also extends.

As before, we consider the hermitian self-dual Ok-lattice L “ HomOkpa0, aq

in V “ HomkpW0,W q. The dual lattice of L with respect to the bilinearform r¨, ¨s is L1 “ δ´1L. Let

SL Ă SpV pAf qq

be the space of Schwartz-Bruhat functions that are supported on pL1 and

invariant under translations by pL.Recall from Remark 2.1.3 the identification

D – tw P εV pCq : rw,ws ă 0uCˆ.We may also identify

D – tnegative definite k-stable R-planes z Ă V pRqu

using the isomorphism V pRq Ñ V pCq εÝÑ εV pCq. For any x P V and z P D,

let xz be the orthogonal projection of x to the plane z Ă V pRq, and let xzKbe the orthogonal projection to zK.

For pτ, z, gq P HˆD ˆGpAf q and ϕ P SL, we define a theta function

θpτ, z, g, ϕq “ÿ

xPV

ϕpg´1xq ¨ ϕ8pτ, z, xq,

where the Schwartz function at 8,

ϕ8pτ, z, xq “ v ¨ e2πiQpxzKqτ`2πiQpxzqτ ,

is the usual Gaussian involving the majorant associated to z. We may viewθ as a function HˆD ˆGpAf q Ñ S_L . As a function in pz, gq it is invariantunder the left action of GpQq. Under the right action of K it satisfies thetransformation law

θpτ, z, gk, ϕq “ θpτ, z, g, ωLpkqϕq, k P K,

where ωL denotes the action of K on SL by the Weil representation andv “ Impτq. In the variable τ P H it transforms as a S_L -valued modular formof weight n´ 2 for SL2pZq.

Fix an f P H82´npD,χq with Fourier expansion as in (7.2.1), and assumethat c`pmq P Z for m ď 0. We associate to f the divisors

ZKrapfq “ÿ

mą0

c`p´mq ¨ ZKrapmq

ZtotKrapfq “

ÿ

mą0

c`p´mq ¨ ZtotKrapmq

on SKra and S˚Kra, respectively. As the actions of SL2pZq and K via the Weilrepresentation commute, the associated SL-valued harmonic Maass form


f is invariant under K. Hence the natural pairing SL ˆ S_L Ñ C gives

rise to a scalar valued function pfpτq, θpτ, z, gqq in the variables pτ, z, gq PHˆD ˆGpAf q, which is invariant under the right action of K and the leftaction of GpQq. Hence it descends to a function on SL2pZqzHˆShpG,DqpCq.

We define the regularized theta lift of f as

Θregpz, g, fq “

ż reg

SL2pZqzH

`

fpτq, θpτ, z, gq˘ du dv

v2.

Here the regularization of the integral is defined as in [Bor98, BF04, BHY15].We extend the incomplete Gamma function

Γp0, tq “

ż 8

te´v

dv

v

to a function on Rě0 by setting

rΓp0, tq “

#

Γp0, tq if t ą 0,

0 if t “ 0.

Theorem 7.2.1. The regularized theta lift Θregpz, g, fq defines a smoothfunction on SKrapCqr ZKrapfqpCq. For g P GpAf q and z0 P D, there existsa neighborhood U Ă D of z0 such that

Θregpz, g, fq ´ÿ

xPgLxKz0

c`p´xx, xyq ¨ rΓ`

0, 4π|xxz, xzy|˘

is a smooth function on U .

Proof. Note that the sum over x P gL X zK0 is finite. Since ShpG,DqpCqdecomposes into a finite disjoint union of connected components of the form

pGpQq X gKg´1qzD,

where g P GpAf q, it suffices to consider the restriction of Θregpfq to thesecomponents.

On such a component, Θregpz, g, fq is a regularized theta lift considered

in [BHY15, Section 4] of the vector valued form f for the lattice

gL “ gpLX V “ HomOkpga0, gaq Ă V,

and hence the assertion follows from (7.2.2) and [BHY15, Theorem 4.1].

Remark 7.2.2. Let ∆D denote the UpV qpRq-invariant Laplacian on D. Thereexists a non-zero real constant c (which only depends on the normalizationof ∆D and which is independent of f), such that

∆DΘregpz, g, fq “ c ¨ degZKrapfqpCq

on the complement of the divisor ZKrapfqpCq.


Using the fact that

Γp0, tq “ ´ logptq ` Γ1p1q ` optq

as tÑ 0, Theorem 7.2.1 implies that Θregpfq is a (sub-harmonic) logarithmicGreen function for the divisor ZKrapfqpCq on the non-compactified Shimuravariety SKrapCq. These properties, together with an integrability condition,characterize it uniquely up to addition of a locally constant function [BHY15,Theorem 4.6]. The following result describes the behavior of Θregpfq on thetoroidal compactification.

Theorem 7.2.3. On S˚KrapCq, the function Θregpfq is a logarithmic Greenfunction for the divisor Ztot

KrapfqpCq with possible additional log-log singular-ities along the boundary in the sense of [BGKK07].

Proof. As in the proof of Theorem 7.2.1 we reduce this to showing thatΘregpfq has the correct growth along the boundary of the connected compo-nents of S˚KrapCq. Then it is a direct consequence of [BHY15, Theorem 4.10]and [BHY15, Corollary 4.12].

Recall that ωan is the tautological bundle on

D –


Cˆ.

We define the Petersson metric ¨ on ωan by

w2 “ ´rw,ws

4πeγ,

where γ “ ´Γ1p1q denotes Euler’s constant. This choice of metric on ωan

induces a metric on the line bundle ω on SKrapCq defined in §2.4, whichextends to a metric over S˚KrapCq with log-log singularities along the bound-ary [BHY15, Proposition 6.3]. We obtain a hermitian line bundle on S˚Kra,denoted

pω “ pω, ¨ q.

If f is actually weakly holomorphic, that is, if it belongs to M !,82´npD,χq,

then Θregpfq is given by the logarithm of a Borcherds product. Moreprecisely, we have the following theorem, which follows immediately from[Bor98, Theorem 13.3] and our construction of ψpfq as the pullback of aBorcherds product, renormalized by (6.2.3), on an orthogonal Shimura va-riety.

Theorem 7.2.4. Let f P M !,82´npD,χq be as in (5.2.2). The Borcherds

product ψpfq of Theorem 5.3.1 satisfies

Θregpfq “ ´ log ψpfq2.


7.3. Generating series of arithmetic special divisors. We can nowdefine arithmetic special divisors on S˚Kra, and prove a modularity result forthe corresponding generating series in the codimension one arithmetic Chowgroup. This result extends Theorem 7.1.5.

Recall our hypothesis that n ą 2, and let m be a positive integer. Thereexists a unique element fm P H82ńpD,χq whose Fourier expansion at thecusp 8 has the form

fm “ q´m Òp1q

as q Ñ 0. According to Theorem 7.2.3, its regularized theta lift Θregpfmq isa logarithmic Green function for Ztot

Krapmq.

Denote by xCh1

QpS˚Kraq the arithmetic Chow group [GS90] of rational equiv-alence classes of arithmetic divisors with Q-coefficients. We allow the Greenfunctions of our arithmetic divisors to have possible additional log-log errorterms along the boundary of S˚KrapCq, as in the theory of [BGKK07]. Form ą 0 define an arithmetic special divisor

pZtotKrapmq “ pZtot

Krapmq,Θregpfmqq P xCh

1

QpS˚Kraq

on S˚Kra, and for m “ 0 set

pZtotKrap0q “ pω´1 P xCh

1

QpS˚Kraq.

Theorem 7.3.1. The formal q-expansion

pφpτq “ÿ

mě0

pZtotKrapmq ¨ q

m P xCh1

QpS˚Kraqrrqss,(7.3.1)

is a holomorphic modular form of level D, weight n, and character χ.

Proof. For any input form f PM !,82ńpD,χq as in (5.2.2), the relation in the

Chow group given by the Borcherds product ψpfq is compatible with theGreen functions, in the sense that

´ log ψpfq2 “ÿ

mą0

cp´mq ¨Θregpfmq.

Indeed, this directly follows from f “ř

mą0 cp´mqfm and Theorem 7.2.4.This observation allows us to simply repeat the argument of Theorem

7.1.5 on the level of arithmetic Chow groups: The arithmetic divisor

xdivpψpfqqdef“

`

divpψpfqq,´ log ψpfq2˘

P xCh1

QpS˚Kraq

satisfies both

xdivpψpfqq “ÿ

mą0

cp´mq ¨ pZtotKrapmq ´

k

2¨ Exc`

ÿ

r|D

γrcrp0qÿ

p|r

S˚KraFp

´ÿ

mą0

cp´mq

2

ÿ

sPπ0pSingq



where the vertical divisors Exc, S˚KraFp, and Excs and are endowed with

trivial Green functions, and

xdivpψpfqq “ÿ

r|D

γrcrp0q ¨ pZtotKrap0q.

Using these relations the proof goes through verbatim.

In the theory of arithmetic Chow groups [BGKK07, GS90] one typicallyworks on a regular scheme such as S˚Kra. However, the codimension onearithmetic Chow group of S˚Pap makes perfect sense: one only needs to spec-ify that it consists of rational equivalence classes of Cartier divisors on S˚Pap

endowed with a Green function.With this in mind one can use the equality Ytot

PappmqpCq “ 2 ¨ZtotKrapmqpCq

in the complex fiber S˚PappCq – S˚KrapCq to define arithmetic divisors

pYtotPappmq “ pYtot

Pappmq, 2 ¨Θregpfmqq P xCh

1

QpS˚Papq

for m ą 0. For m “ 0 we define

pYtotPapp0q “

pΩ´1 P xCh1

QpS˚Papq,

where the metric on Ω is induced from that on ω, again using Ω – ω2 inthe complex fiber. Repeating the proof of Theorem 7.1.4 on the level ofarithmetic Chow groups proves that the formal q-expansion

ÿ

mě0

pYtotPappmq ¨ q

m P xCh1

QpS˚Papqrrqss,

is a modular form of level D, weight n, and character χ.

7.4. Non-holomorphic generating series of special divisors. In thissubsection we discuss a non-holomorphic variant of the generating series(7.3.1), which is obtained by endowing the special divisors with other Greenfunctions, namely with those constructed in [How12, How15] following themethod of [Kud97b]. By combining Theorem 7.3.1 with a recent result ofEhlen and Sankaran [ES16], we show that the non-holomorphic generatingseries is also modular.

For every m P Z and v P Rą0 define a divisor

BKrapm, vq “1

4πv

ÿ

Φ

#tx P L0 : xx, xy “ mu ¨ S˚KrapΦq

with real coefficients on S˚Kra. Here the sum is over all K-equivalence classesof proper cusp label representatives Φ in the sense of §3.2, L0 is the hermitianOk-module of signature pn ´ 2, 0q defined by (3.1.4), and S˚KrapΦq is theboundary divisor of Theorem 3.7.1. Note that BKrapm, vq is trivial for allm ă 0. We define special divisors in Ch1

RpS˚Kraq depending on the parameter


v by putting

ZtotKrapm, vq “

#

Z˚Krapmq ` BKrapm, vq if m ‰ 0

ω´1 ` BKrap0, vq if m “ 0.

Following [How12, How15, Kud97b], Green functions for these divisorscan be constructed as follows. For x P V pRq and z P D we put

Rpx, zq “ ´2Qpxzq,

and we let

βpvq “

ż 8

1e´vt

dt

t.

For m P Z and pv, z, gq P Rą0 ˆD ˆGpAf q, we define a Green function

Ξpm, v, z, gq “ÿ

xPV rt0uQpxq“m

χpLpg´1xq ¨ βp2πvRpx, zqq,(7.4.1)

where χpLP SL denotes the characteristic function of pL. As a function of the

variable pz, gq, (7.4.1) is invariant under the left action of GpQq and underthe right action of K, and so descends to a function on Rą0 ˆ ShpG,DqpCq.It was proved in [How15, Theorem 3.4.7] that Ξpm, vq is a logarithmic Greenfunction for Ztot

Krapm, vq when m ‰ 0. When m “ 0 it is a logarithmic Greenfunction for BKrap0, vq.

Consequently, we obtain arithmetic special divisors in xCh1

RpS˚Kraq depend-ing on the parameter v by putting

pZtotKrapm, vq “

#`

ZtotKrapm, vq,Ξpm, vq

˘

if m ‰ 0

pω´1 ` pBKrap0, vq,Ξp0, vqq ´ p0, log vq if m “ 0.

Note that for m ă 0 these divisors are entirely supported in the archimedianfiber.

Theorem 7.4.1. The formal q-expansion

pφnon-holpτq “ÿ

mPZ

pZtotKrapm, vq ¨ q

m P xCh1

RpS˚Kraqrrqss,

is a non-holomorphic modular form of level D, weight n, and character χ.Here q “ e2πiτ and v “ Impτq.

Here the assertion about modularity is to be understood as in [ES16,Definition 4.11]. In our situation it reduces to the statement that there is asmooth function spτ, z, gq on HˆShpG,DqpCq with the following properties:

(1) in pz, gq the function spτ, z, gq has at worst log-log-singularities atthe boundary of ShpG,DqpCq (in particular it is a Green functionfor the trivial divisor);

(2) spτ, z, gq transforms in τ as a non-holomorphic modular form of levelD, weight n, and character χ;


(3) the difference pφnon-holpτq ´ spτ, z, gq belongs to the space

MnpD,χq bC xCh1

CpS˚Kraq ‘ pRn´2Mn´2pD,χqq bC xCh1

CpS˚Kraq,

where Rn´2 denotes the Maass raising operator as in Section 8.4.

Proof. Theorem 4.13 of [ES16] states that the difference pφnon-holpτq´ pφpτq isa non-holomorphic modular form of level D, weight n, and character χ with

values in xCh1

CpS˚Kraq. Hence the assertion follows from Theorem 7.3.1.

Remark 7.4.2. According to Theorem 4.18 of [ES16], the difference pφpτq ´pφnon-holpτq has trivial holomorphic projection. Hence the two generating

series pφnon-holpτq and pφpτq define the same arithmetic theta lifts on SnpD,χq.

8. Appendix: some technical calculations

We collect some technical arguments and calculations. Strictly speaking,none of these are essential to the proofs in the body of the text. We explainthe connection between the fourth roots of unity γp defined by (5.3.1) andthe local Weil indices appearing in the theory of the Weil representation,provide alternative proofs of Propositions 6.1.2 and 6.3.3, and explain ingreater detail how Proposition 6.3.1 is deduced from the formulas of [Kud16].

8.1. Local Weil indices. In this subsection, we explain how the quantityγp defined in (5.3.1) is related to the local Weil representation.

Let L Ă V be as in §6.1. Recall that SL “ CrL1Ls is identified with asubspace SL Ă SpV pAf qq, via µ P L1L ÞÑ φµ, the characteristic function

of µ ` pL. Since dimQ V “ 2n and since D is odd, the representation ωL ofSL2pZq on SL is the pullback via

SL2pZq ÝÑź

p|D

SL2pZpq

of the representation

ωL “â

p|D

ωp,

where ωp “ ωLp is the Weil representation of SL2pZpq on SLp Ă SpVpq.These Weil representations are defined using the standard global additive

character ψ “ bpψp which is trivial on pZ and on Q and whose restriction toR Ă A is given by ψpxq “ epxq. Recall that, for a P Qˆp and b P Qp

ωppnpbqqφpxq “ ψppbQpxqq ¨ φpxq

ωppmpaqqφpxq “ χnk,ppaq ¨ |a|np ¨ φpaxq

ωppwqφpxq “ γp

ż

Vp

ψpp´rx, ysq ¨ φpyq dy, w “

ˆ

´11

˙

,


where γp “ γppLq is the Weil index of the quadratic space Vp with respectto ψp and χk,p is the quadratic character of Qˆp corresponding to kp. Notethat dy is the self-dual measure with respect to the pairing ψpprx, ysq.

Lemma 8.1.1.

(1) When restricted to the subspace SLp Ă SpVpq, the action of γ PSL2pZpq depends only on the image of γ in SL2pFpq.

(2) The Weil index is given by

γp “ εńp ¨ pD, pqnp ¨ invppVpq

where pa, bqp is the Hilbert symbol for Qp and invppVpq is the invari-ant of Vp in the sense of (1.7.2).

Proof. (i) It suffices to check this on the generators. We omit this.(ii) We can choose an Ok,p-basis for Lp such that the matrix for the hermitianform is diagpa1, . . . , anq, with aj P Zˆp . The matrix for the bilinear formrx, ys “ TrKpQppxx, yyq is then diagp2a1, . . . , 2an, 2Da1, . . . , 2Danq. Then,according to the formula for βV in [Kud94, p. 379], we have

γ´1p “ γQpp

1

2¨ ψp ˝ V q “

nź

j“1

γQppajψpq ¨ γQppDajψpq,

where we note that, in the notation there, xpwq “ 1, and j “ jpwq “ 1.Next by Proposition A.11 of the Appendix to [RR93], for any α P Zˆp , wehave γQppαψpq “ 1 and

γQppαpψpq “

ˆ

´α

p

˙

¨ εp “ p´α, pqp ¨ εp.

Here note that if η “ αpψp, then the resulting character η of Fp is given by

ηpaq “ ψppp´1aq “ ep´p´1aq.

and γFppηq “´

´1p

¯

¨ εp. Thus

γp “ εńp ¨ p´Dp, pqnp ¨ pdetpV q, pqp,

as claimed.

8.2. A direct proof of Proposition 6.1.2. The proof of Proposition 6.1.2,which expresses the Fourier coefficients of the vector valued form f in termsof those of the scalar valued form f P M !

2ńpD,χq, appealed to the moregeneral results of [Sch09]. In some respects, it is easier to prove Proposi-tion 6.1.2 from scratch than it is to extract it from [loc. cit.]. This is whatwe do here.

Recall that f is defined from f by the induction procedure of (6.1.1), andthat the coefficients cpm,µq in its Fourier expansion (6.1.2) are indexed bym P Q and µ P L1L. Recall that, for r | D, rs “ D,

Wr “

ˆ

rα βDγ rδ

˙

“ Rr

ˆ

r1

˙

, Rr “

ˆ

α βsγ rδ

˙

P Γ0psq.


Note that

(8.2.1) Γ0pDqzSL2pZq “ Γ0pDqzSL2pZqΓpDq »ź

p|D

BpzSL2pFpq,

so this set has orderś

p|Dpp` 1q. A set of coset representatives is given by

ğ

r|Dc pmod rq

Rr

ˆ

1 c1

˙

.

Now, using (3.3.1), we haveˆ

fˇ

ˇ

2ńRr

ˆ

1 c1

˙˙

pτq “

ˆ

fˇ

ˇ

2ńWr

ˆ

r´1 r´1c1

˙˙

pτq(8.2.2)

“ χrpβqχspαqÿ

m"´8

rn2´1crpmq ¨ e

2πimpτ`cqr

On the other hand, the image of the inverse of our coset representative onthe right side of (8.2.1) has components

$

’

’

’

’

&

’

’

’

’

%

˜

1 ć

1

¸˜

0 ´β

´sγ α

¸

if p | r

˜

1 ć

1

¸˜

rδ ´β

0 α

¸

if p | s.

Note that rαδ ´ sβγ “ 1. Then, as elements of SL2pFpq, we have$

’

’

’

’

&

’

’

’

’

%

˜

1 ć

1

¸˜

β

β´1

¸˜

´1

1

¸˜

1 αβ

1

¸

if p | r

˜

1 ć

1

¸˜

α´1

0 α

¸˜

1 ´αβ

1

¸

if p | s.

The element on the second line just multiplies φ0,p by χppαq. For the elementon the first line, the factor on the right fixes φ0 and

ωp

ˆˆ

´11

˙˙

φ0 “ γp pń

2

ÿ

µPL1pLp

φµ.

Thus, the element on the first line carries φ0,p to

χppβqγp pń

2

ÿ

µPL1pLp

ψppćQpµqqφµ.

Recall from (6.1.3) that for µ P L1L, rµ is the product of the primes p | Dsuch that µp ‰ 0. Thus

(8.2.3) ωL

ˆ

Rr

ˆ

1 c1

˙˙´1

φ0 “ χspαqχrpβq γr rń

2

ÿ

µPL1Lrµ|r

e2πicQpµqφµ.


Taking the product of (8.2.2) and (8.2.3) and summing on c and on r, weobtain

ÿ

r|D

γr ¨ r´1

ÿ

c pmod rq

ÿ

µPL1Lrµ|r

e2πicQpµqφµÿ

m"´8

crpmqe2πimpτ`cq

r

“ÿ

r|D

γrÿ

µPL1Lrµ|r

φµÿ

m"´8mr`QpµqPZ

crpmq qmr

“ÿ

mPQm"´8

ÿ

µPL1Lm`QpµqPZ

ÿ

rrµ|r|D

γrcrpmrqφµ qm

This gives the claimed general expression for cpm,µq and completes the proofof Proposition 6.1.2.

8.3. A more detailed proof of Proposition 6.3.1. In this section, weexplain in more detail how to obtain the product formula of Proposition 6.3.1from the general formula given in [Kud16].

For our weakly holomorphic SL-valued modular form f of weight 2 ´ n,with Fourier expansion given by (6.1.2), the corresponding meromorphic

Borcherds form Ψpfq on D` has a product formula [Kud16, Corollary 2.3]in a neighborhood of the 1-dimensional boundary component associated toL´1. It is given as a product of 4 factors, labeled (a), (b), (c) and (d).We note that, in our present case, there is a basic simplification in factor(b) due to the restriction on the support of the Fourier coefficients of f .More precisely, for m ą 0, cp´m,µq “ 0 for µ R L, and cp´m, 0q “ cp´mq.In particular, if x P L1 with rx, e´1s “ rx, f´1s “ 0, then Qpxq “ Qpx0q,where x0 is the pL0qQ component of x. If x0 ‰ 0, then Qpxq ą 0, and

cp´Qpxq, µq “ 0 for µ R L. The factors for Ψpfq are then given by:(a)

ź

xPL1rx,f´1s“0rx,e´1są0

mod LXQ f´1

`

1´ e´2πirx,ws˘cp´Qpxq,xq

.

(b)

P1pw0, τ1qdef“

ź

xPL0rx,W0są0

ˆ

ϑ1p´rx,ws, τ1q

ηpτ1q

˙cp´Qpxqq

,

where W0 is a Weyl chamber in V0pRq, as in [Kud16, §2].(c)

P0pτ1qdef“

ź

xPδ´1L´1L´1x‰0

ˆ

ϑ1p´rx,ws, τ1q

ηpτ1qeπirx,ws¨rx,e1s

˙cp0,xq2


(d) and

κ ηpτ1qcp0,0q qI02 ,

where κ is a scalar of absolute value 1, and

I0 “ ´ÿ

m

ÿ

xPL1XpL´1qK

mod L´1

cp´m,xqσ1pm´Qpxqq.

The factors given in Proposition 6.3.1 are for the form

ψgpfqdef“ p2πiqcp0,0qΨp2fq

The quantity q2 in [Kud16] is our epξq, and τ1 there is our τ .Recall from (3.9.2) that δ´1L´1 “ Ze´1`D

´1Zf´1, so that, in factor (c),the product runs over vectors D´1b f´1, with b mod D, b ‰ 0. For thesevectors rx, e1s “ 0. In the formula for I, x runs over vectors of the form

x “ ´b

Df´1 ` x0,

with x0 P δ´1L0. But, again, if x0 ‰ 0, Qpxq “ Qpx0q ą 0 and cp´Qpxq, xq “

0 unless b “ 0, and so the sum in that term runs over x0 P L0 x0 ‰ 0 andover ´ b

D f´1’s.

Thus the factors for ψgpfq are given by:(a)

ź

xPL1rx,f´1s“0rx,e´1są0

mod LXQ f´1

`

1´ e´2πirx,ws˘2 cp´Qpxq,xq

,

(b)

P1pw0, τ1qdef“

ź

x0PL0x0‰0

ˆ

ϑ1p´rx0, ws, τ1q

ηpτ1q

˙cp´Qpx0qq

,

(c)

P0pτ1qdef“

ź

bPZDZb‰0

ˆ

ϑ1p´rx,ws, τ1q

ηpτ1q

˙cp0, bDf´1q

,

(d) and, setting k “ cp0, 0q,

κ2 p 2πi η2pτqqk q2I02 ,

where κ is a scalar of absolute value 1, and

I0 “ ´2ÿ

mą0

ÿ

x0PL0

cp´mqσ1pm´Qpx0qq `1

12

ÿ

bPZDZcp0,

b

Df´1q.

Here note that for ψgpfq “ p2πiqcp0,0qΨp2fq we have multiplied the previous

expression by 2.


Finally recall

w “ ´ξe´1 ` pτξ ´Qpw0qqf´1 ` w0 ` τe1 ` f1.

If rx, f´1s “ 0, then x has the form

x “ áe´1 ´b

Df´1 ` x0 ` ce1,

so that

rx,ws “ ć ξ ` rx0, w0s ´ aτ ´b

D,

and

Qpxq “ ác`Qpx0q.

Using these values, the formulas given in Proposition 6.3.1 follow immedi-ately.

8.4. A direct proof of Proposition 6.3.3. Here we give a direct proofof Proposition 6.3.3, which does not rely on Corollary 6.3.2. We begin byrecalling some general facts about derivatives of modular forms.

We let q ddq be the Ramanujan theta operator on q-series. Recall that the

image under q ddq of a holomorphic modular form g of weight k is in general

not a modular form. However, the function

Dpgq “ qd

dqpgq ´

k

12gE2(8.4.1)

is a holomorphic modular form of weight k`2 (see e.g. [BHY], Section 4.2).Here

E2pτq “ ´24ÿ

mě0

σ1pmqqm

denotes the non-modular Eisenstein series of weight 2 for SL2pZq. In partic-ular σ1p0q “ ´

124 . We extend σ1 to rational arguments by putting σ1prq “ 0

if r R Zě0. If Rk “ 2i BBτ `

kv denotes the Maass raising operator and

E˚2 pτq “ E2pτq´3πv the non-holomorphic (but modular) Eisenstein series of

weight 2, we also have

Dpgq “ ´1

4πRkpgq ´

k

12gE˚2 .

Proposition 8.4.1. Let f PM !,82ńpD,χq as in (5.2.2). The integer

I “1

12

ÿ

αPδ´1L´1L´1

cp0, αq ´ 2ÿ

mą0

cp´mqÿ

xPL0

σ1pm´Qpxqq.

defined in Proposition 6.3.1 is equal to the integer

multΦpfq “1

n´ 2

ÿ

xPL0

cp´QpxqqQpxq

defined by (5.2.4).


Proof. Consider the S_L0-valued theta function

Θ0pτq “ÿ

xPL10

qQpxqχ_x`L0PMn´2pω

_L0q.

Applying the above construction (8.4.1) to Θ0 we obtain an S_L0-valued

modular form

DpΘ0q “ÿ

xPL10

QpxqqQpxqχ_x`L0ń´ 2

12Θ0E

˚2 PMnpω

_L0q

of weight n. For its Fourier coefficients we have

DpΘ0q “ÿ

νPL10L0

ÿ

mě0

bpm, νqqmχ_ν

bpm, νq “ÿ

xPν`L0Qpxq“m

Qpxq ` 2pn´ 2qÿ

xPν`L0

σ1pm´Qpxqq.

As in [BHY15, (4.8)], an SL-valued modular form F induces an SL0-valued form FL0 . If we denote by Fµ the components of F with respect tothe standard basis pχµq of SL, we have

FL0,ν “ÿ

αPδ´1L´1L´1

Fν`α(8.4.2)

for ν P L10L0.

Let f PM !2ńpωLq be the SL-valued form corresponding to f , as in (6.1.1).

Using (8.4.2) we obtain

fL0 PM!2ńpωL0q

with Fourier expansion

fL0 “ÿ

ν,m

ÿ

αPδ´1II

cpm, ν ` αqqmχν`L0 .

We consider the natural pairing between the SL0-valued modular form fL0

of weight 2´ n and the S_L0-valued modular form DpΘ0q of weight n,

pfL0 , DpΘ0qq PM!2pSL2pZqq.

By the residue theorem, the constant term of the q-expansion vanishes, thatis,

ÿ

mě0

ÿ

νPL10L0

αPδ´1II

cp´m, ν ` αqbpm, νq “ 0.(8.4.3)

We split this up in the sum over m ą 0 and the contribution from m “ 0.Employing Proposition 6.1.2, we obtain that the sum over m ą 0 is equal to

ÿ

mą0

cp´mqbpm, 0q.


For the contribution of m “ 0 we notice

bp0, νq “

#

ń´212 , ν “ 0 P L10L0,

0, ν ‰ 0.

Hence this part is equal to

ń´ 2

12

ÿ

αPδ´1L´1L´1

cp0, αq.

Inserting the two contributions into (8.4.3), we obtain

0 “ÿ

mą0

cp´mqbpm, 0q ń´ 2

12

ÿ

αPδ´1L´1L´1

cp0, αq

“ÿ

mą0

cp´mq

ˆ

ÿ

xPL0Qpxq“m

Qpxq ` 2pn´ 2qÿ

xPL0

σ1pm´Qpxqq

˙

ń´ 2

12

ÿ

αPδ´1L´1L´1

cp0, αq

“ÿ

xPL0

cp´QpxqqQpxq ` 2pn´ 2qÿ

mą0

cp´mqÿ

xPL0

σ1pm´Qpxqq

ń´ 2

12

ÿ

αPδ´1L´1L´1

cp0, αq

“ pn´ 2qmultΦpfq ´ pn´ 2qI.

This concludes the proof of the proposition.

Now we verify directly the other claim of Proposition 6.3.3: the function

P1pτ, w0q “ź

mą0

ź

xPL0Qpxq“m

Θ`

τ, xw0, xy˘cp´mq

satisfies the transformation law (3.9.10) with respect to the translation ac-tion of bL0 on the variable w0.

First recall that, for a, b P Z,

Θpτ, z ` aτ ` bq “ ep1

2a2τ ` az `

1

2pa´ bqq´1 Θpτ, zq.

If we write α “ aτ ` b and τ “ u` iv, then

a “Impαq

v“α´ α

2iv, b “ Repαq ´

u

vImpαq.

Then

1

2a2τ ` az `

1

2pa´ bq “

1

4ivpα´ αqα`

1

2ivpα´ αqz `

1

2pa´ b´ abq.


For z and w in C, write

Rpz, wq “ Rτ pz, wq “ Bτ pz, wq ´Hτ pz, wq “1

vzpw ´ wq.

Then1

4vpα´ αqα`

1

2vpα´ αqz “

1

2Rpz, αq `

1

4Rpα, αq,

and we can write

Θpτ, z ` αq “ expp´πRpz, αq ´π

2Rpα, αqq ¨ exppπipa´ b´ abqq´1 Θpτ, zq.

We will consider the contribution of the 12pa´ b´ abq term separately.

For β P V0, we have xw0 ` β, xy “ xw0, xy ` xβ, xy. Suppose that for allx P L0, we have xβ, xy “ aτ ` b for a and b in Z. Writing b “ Z` Zτ , thisis precisely the condition that β P bL0. Then we obtain a factor

exp

¨

˚

˚

˝

´πÿ

mą0

ÿ

xPL0Qpxq“m

cp´mq

«

R`

xw0, xy, xβ, xy˘

`R`

xβ, xy, xβ, xy˘

2

ff

˛

‹

‹

‚

.

Expanding the sum and using the hermitian version of Borcherds’ quadraticidentity from the proof of Proposition 5.2.2, we have

ÿ

xPL0

cp´Qpxqq

v

„

xw0, xyxβ, xy ´ xw0, xyxx, βy `xβ, xyxβ, xy

2´xβ, xyxx, βy

2

“ ´1

v

ˆ

xw0, βy `1

2xβ, βy

˙

¨1

2n´ 4¨ÿ

xPL0

cp´Qpxqq rx, xs

“ ´1

v

ˆ

xw0, βy `1

2xβ, βy

˙

¨multΦpfq.

Thus, using I “ multΦpfq, we have a contribution of

exp´πxw0, βy

v`πxβ, βy

2v

¯I

to the transformation law.Next we consider the quantity

a´ b´ ab

“Impαq

v´ Repαq ´

u Impαq

v´

Impαq

v

ˆ

Repαq ú Impαq

v

˙

“α´ α

2iv´pα` αq

2úpα´ αq

2iv´α´ α

2iv

ˆ

pα` αq

2úpα´ αq

2iv

˙

.

This will contribute expp´πiAq, where A is defined as the sum

ÿ

x‰0

cp´Qpxqq

„

α´ α

2iv´α` α

2úpα´ αq

2iv´α´ α

2iv

ˆ

pα` αq

2úpα´ αq

2iv

˙


where α “ xβ, xy. Since x and ´x both occur in the sum, the linear termsvanish and

A “ÿ

x‰0

cp´Qpxqq

„

´α´ α

2iv

ˆ

pα` αq

2´upα´ αq

2iv

˙

.

Using the hermitian version of Borcherds quadratic identity, as in the proofof Proposition 5.2.2, we obtain

A “uI

2v2¨ xβ, βy.

Thus we have

P1pτ, w0 ` βq

“ P1pτ, w0q ¨ exp´π

vxw0, βy `

π

2vxβ, βy

¯I¨ exp

´

´2πiuxβ, βy

4v2

¯I.

Finally, we recall the conjugate linear isomorphism L´1 – b of (3.9.7)defined by e´1 ÞÑ τ and f´1 ÞÑ 1. As

δ´1L´1 “ Ze´1 `D´1Zf´1,

we have ´δ´1τ “ aτ `D´1b for some a, b P Z, and hence

τ “ ´D´1bpa` δ´1q´1.

This gives uv “ aD12 . Also, using

δe´1 “ ´Dae´1 ´ b f´1,

we have

1

2p1` δq e´1 “

1

2p1´Daq e´1 ´

1

2b f´1 P Ze´1 ` Zf´1 “ L´1.

Thus a is odd and b is even. Recall that Npbq “ 2v?D. Thus

u

4v2“

aD12

2NpbqD12

,

and, since xβ, βy P Npbq, we have

exp´

´2πiuxβ, βy

4v2

¯

“ exp´

´πixβ, βy

Npbq

¯

“ ˘1.

The transformation law is then

P1pτ, w0 ` βq “ exp´π

vxw0, βy `

π

2vxβ, βy ´ iπ

xβ, βy

Npbq

¯I¨ P1pτ, w0q,

as claimed in Proposition 6.3.3.


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Fachbereich Mathematik, Technische Universitat Darmstadt, Schlossgarten-strasse 7, D–64289 Darmstadt, Germany

E-mail address: [email protected]

Department of Mathematics, Boston College, Chestnut Hill, MA 02467,USA



Department of Mathematics, University of Toronto, 40 St. George St.,BA6290, Toronto, ON M5S 2E4, Canada


Mathematisches Institut der Universitat Bonn, Endenicher Allee 60, 53115Bonn, Germany


Department of Mathematics, University of Wisconsin Madison, Van VleckHall, Madison, WI 53706, USA


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